SlideShare a Scribd company logo

53 multiplication and division of rational expressions

Download as PPTX, PDF
A
alg1testreview

The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.

Education
1 of 88
Downloaded 14 times
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
Multiplication and Division of Rational Expressions Frank Ma © 2011
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) *
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 )
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) (x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) = (x + 3)(x + 2) x * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2) In the next section, we meet the following type of problems.
Multiplication and Division of Rational Expressions Example B. Simplify and expand the answers. a. x + 3 x – 1 (x2 – 1)
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions Example B. Simplify and expand the answers. a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1)
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) – x + 1 (x – 3)(x + 1) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) = (x – 2)(x + 1) + (–x –1)(x + 3) Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) = (x – 2)(x + 1) + (–x –1)(x + 3) = x2 – x – 2 – x2 – 4x – 3 Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions Example B. Simplify and expand the answers. a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) = (x – 2)(x + 1) + (–x –1)(x + 3) = x2 – x – 2 – x2 – 4x – 3 = –5x – 5
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate We convert division by an expression of multiplying by its reciprocal.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate Example C. Simplify (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) * (9 – x2) We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) * (–1) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) * (–1) = –2(x – 1) (3 + x) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Multiplication and Division of Rational Expressions Besides the expanded form and factored forms, rational expressions may also be split into sums or differences.
Multiplication and Division of Rational Expressions Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this.
Multiplication and Division of Rational Expressions Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (2x – 6) 3x2 b. =
Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. =
Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2
Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2 = – 2 x2 2 3x
Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2 = – 2 x2 2 3x II. Long Division Long division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable.
Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2 = – 2 x2 2 3x II. Long Division Long division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable. Specifically, long division gives relevant results only when the degree of the numerator is the same or more than the degree of the denominator.
Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient.
Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 1
Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 1 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45
Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 1 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered.
Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 15 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered. 40 5 Let’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 15 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: N D = Q + R D and that R (the remainder) is smaller then D (no more quotient). 40 5 where Q is the quotient Let’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 15 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: N D = Q + R D and that R (the remainder) is smaller then D (no more quotient). 40 5 125 8 = 15 + 5 8 where Q is the quotient Let’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational Expressions Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Make sure the terms are in order. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Make sure the terms are in order. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. enter the quotients of the leading terms 2x/x = 2 Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. enter the quotients of the leading terms 2x/x = 2 Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 2x + 6 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 –) i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. –) i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 –) Stop. No more quotient since x can’t going into 12. iii. Repeat steps i and ii until no more quotient may be entered. i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) Hence we may write (2x – 6) (x + 3) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) Hence we may write (2x – 6) (x + 3) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Q R Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: = 2 – 12 x + 3 –) Hence we may write (2x – 6) (x + 3) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Q R Q R Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions Example F. Divide using long division ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 x – 2 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Multiplication and Division of Rational Expressions )x + 3 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 Make sure the terms are in order. N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27–) 30 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient –) x2 – 6x + 3 –9x – 27–) 30 Stop. No more quotient since x can’t going into 30. Hence 30 is the remainder. and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27–) 30 Hence x2 – 6x + 3 x – 2 = N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27–) 30 Hence x2 – 6x + 3 x – 2 = x – 9 + 30 x + 3 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R Example F. Divide using long divisionx2 – 6x + 3 x – 2
Ex A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions 1. 10x * 2 5x3 15x 4 * 16 25x4 10x * 35x32. 5. 10 9x4 * 18 5x3 6. 3.12x6* 5 6x14 56x6 27 * 63 8x5 10x * 35x34. 7. 75x 49 * 42 25x3 8. 9. 2x – 4 2x + 4 5x + 10 3x – 6 10. 6 – 4x 3x – 2 x – 2 2x + 4 11. 9x – 12 2x – 4 2 – x 8 – 6x 12. x + 4 –x – 4 4 – x x – 4 13. 3x – 9 15x – 5 3 – x 5 – 15x 14. 42 – 6x –2x + 14 4 – 2x –7x + 14 * * * * * * 15. (x2 + x – 2 ) (x – 2) (x2 – x) (x2 – 4 ) * 16. (x2 + 2x – 3 ) (x2 – 9) (x2 – x – 2 ) (x2 – 2x – 3) * 17. (x2 – x – 2 ) (x2 – 1) (x2 + 2x + 1) (x2 + x ) * 18. (x2 + 5x – 6 ) (x2 + 5x + 6) (x2 – 5x – 6 ) (x2 – 5x + 6) * 19. (x2 – 3x – 4 ) (x2 – 1) (x2 – 2x – 8) (x2 – 3x + 2) * 20. (– x2 + 6 – x ) (x2 + 5x + 6) (x2 – x – 12 ) (6 – x2 – x) *
Ex. A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions 21. (2x2 + x – 1 ) (1 – 2x) (4x2 – 1) (2x2 – x ) 22. (3x2 – 2x – 1) (1 – 9x2) (x2 + x – 2 ) (x2 + 4x + 4) 23.(3x2 – x – 2) (x2 – x + 2) (3x2 + 4x + 1) (–x – 3x2) 24. (x + 1 – 6x2) (–x2 – 4) (2x2 + x – 1 ) (x2 – 5x – 6) 25. (x3 – 4x) (–x2 + 4x – 4) (x2 + 2) (–x + 2) 26. (–x3 + 9x ) (x2 + 6x + 9) (x2 + 3x) (–3x2 – 9x) Ex. B. Multiply, expand and simplify the results. ÷ ÷ ÷ ÷ ÷ ÷ 27. x + 3 x + 1 (x2 – 1) 28. x – 3 x – 2 (x2 – 4) 29. 2x + 3 1 – x (x2 – 1) 30. 3 – 2x x + 2 (x + 2)(x +1) 31. 3 – 2x 2x – 1 (3x + 2)(1 – 2x) 32. x – 2 x – 3 ( + x + 1 x + 3 )( x – 3)(x + 3) 33. 2x – 1 x + 2 ( – x + 2 2x – 3 ) ( 2x – 3)(x + 2)
Multiplication and Division of Rational Expressions 38. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) 39. x + 3 x2 – 4 ( – 2x + 1 x2 + x – 2 ) ( x – 2)(x + 2)(x – 1) 40. x – 1 x2 – x – 6 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 2)(x + 1) 41. x + 2 x2 – 4x +3 ( – 2x + 1 x2 + 2x – 3 )( x – 3)(x + 3)(x – 1) 34. 4 – x x – 3 ( – x – 1 2x + 3 )( x – 3)(2x + 3) 35. 3 – x x + 2 ( – 2x + 3 x – 3 )(x – 3)(x + 2) Ex B. Multiply, expand and simplify the results. 36. 3 – 4x x + 1 ( – 1 – 2x x + 3 )( x + 3)(x + 1) 37. 5x – 7 x + 5 ( – 4 – 5x x – 3 )(x – 3)(x + 5)
Ex. C. Break up the following expressions as sums and differences of fractions. 42. 43. 44. 45. 46. 47. x2 + 4x – 6 2x2x2 – 4 x2 12x3 – 9x2 + 6x 3x x2 – 4 2x x x8 – x6 – x4 x2 x8 – x6 – x4 Ex D. Use long division and write each rational expression in the form of Q + . R D (x2 + x – 2 ) (x – 1) (3x2 – 3x – 2 ) (x + 2) 2x + 6 x + 2 48. 3x – 5 x – 2 49. 4x + 3 x – 1 50. 5x – 4 x – 3 51. 3x + 8 2 – x 52. –4x – 5 1 – x 53. 54. (2x2 + x – 3 ) (x – 2) 55. 56. (–x2 + 4x – 3 ) (x – 3) (5x2 – 1 ) (x – 4) 57. (4x2 + 2 ) (x + 3) 58. 59. Multiplication and Division of Rational Expressions

More Related Content

PPTX
5 6 substitution and factoring formulas
math123a
 
PPTX
2 2 addition and subtraction ii
math123b
 
PPTX
4 6multiplication formulas
math123a
 
PPTX
43literal equations
alg1testreview
 
PPTX
2 4 solving rational equations
math123b
 
PPTX
3 multiplication and division of rational expressions x
Tzenma
 
PPTX
2.1 reviews of exponents and the power functions
math123c
 
PPTX
44 exponents
alg1testreview
 
5 6 substitution and factoring formulas
math123a
 
2 2 addition and subtraction ii
math123b
 
4 6multiplication formulas
math123a
 
43literal equations
alg1testreview
 
2 4 solving rational equations
math123b
 
3 multiplication and division of rational expressions x
Tzenma
 
2.1 reviews of exponents and the power functions
math123c
 
44 exponents
alg1testreview
 

What's hot (20)

PPTX
4 5 fractional exponents
math123b
 
PPTX
2 1 addition and subtraction i
math123b
 
PPTX
4 3 algebra of radicals
math123b
 
PPTX
1 4 cancellation
math123b
 
PPT
2.5 calculation with log and exp
math123c
 
PPTX
Difference quotient algebra
math260
 
PPTX
4.2 exponential functions and compound interests
math260
 
PPTX
4 4 more on algebra of radicals
math123b
 
PPTX
5 4 equations that may be reduced to quadratics
math123b
 
PPTX
2 6 complex fractions
math123b
 
PPTX
1.3 solving equations
math260
 
PPTX
1exponents
math123a
 
PPTX
1.3 rational expressions
math123b
 
PPTX
4 2 rules of radicals
math123b
 
PPTX
4 1exponents
math123a
 
PPT
3.2 more on log and exponential equations
math123c
 
PPTX
1 2 2nd-degree equation and word problems-x
math123b
 
PPTX
12 rational expressions
elem-alg-sample
 
PPTX
46polynomial expressions
alg1testreview
 
PPTX
4.5 calculation with log and exp
math260
 
4 5 fractional exponents
math123b
 
2 1 addition and subtraction i
math123b
 
4 3 algebra of radicals
math123b
 
1 4 cancellation
math123b
 
2.5 calculation with log and exp
math123c
 
Difference quotient algebra
math260
 
4.2 exponential functions and compound interests
math260
 
4 4 more on algebra of radicals
math123b
 
5 4 equations that may be reduced to quadratics
math123b
 
2 6 complex fractions
math123b
 
1.3 solving equations
math260
 
1exponents
math123a
 
1.3 rational expressions
math123b
 
4 2 rules of radicals
math123b
 
4 1exponents
math123a
 
3.2 more on log and exponential equations
math123c
 
1 2 2nd-degree equation and word problems-x
math123b
 
12 rational expressions
elem-alg-sample
 
46polynomial expressions
alg1testreview
 
4.5 calculation with log and exp
math260
 
Ad

Viewers also liked (9)

PPTX
English v
Bryan Mejia
 
DOC
Respuestas diapositiva 13 factorizacion de polinomios
flor2510
 
PPTX
54 the least common multiple
alg1testreview
 
PDF
AAM CSTX 10 Hunter Ligger, CA
Donald Moore
 
PPTX
peru
maria sanchez
 
PPTX
New Metrics for Marketing
Jennifer Edgerly
 
PPTX
Gafas 3D
Heydi94
 
PPTX
Ppt on karyotyping, chromosome banding and chromosome painting.
ICRISAT
 
PPTX
Aprendizaje autónomo-y-significativo
Ilse Paola Sánchez Ruiz
 
English v
Bryan Mejia
 
Respuestas diapositiva 13 factorizacion de polinomios
flor2510
 
54 the least common multiple
alg1testreview
 
AAM CSTX 10 Hunter Ligger, CA
Donald Moore
 
peru
maria sanchez
 
New Metrics for Marketing
Jennifer Edgerly
 
Gafas 3D
Heydi94
 
Ppt on karyotyping, chromosome banding and chromosome painting.
ICRISAT
 
Aprendizaje autónomo-y-significativo
Ilse Paola Sánchez Ruiz
 
Ad

Similar to 53 multiplication and division of rational expressions (20)

PPTX
13 multiplication and division of rational expressions
elem-alg-sample
 
PDF
Chapter-5-Quadratic-expressions-and-equations.pdf
SueWei1
 
PDF
Expresiones algebraicas pdf
MineidaSuarez
 
PPT
Factoring polynomials
Mark Ryder
 
PPTX
Algebra Expressions and Exponents Math Presentation in Pink Purple Graphic St...
EdlynMalangPanti
 
PPTX
Factoring2
MartinGeraldine
 
PPT
P6 factoring
salamhello
 
PPT
P6 factoring
salamhello
 
PPT
Polynomials
Mark Ryder
 
PDF
Em01 ba
nahomyitbarek
 
PPTX
Factorization of algebraic expressions math
deepanoel
 
PPTX
1.2 algebraic expressions t
math260
 
PPT
Algebra Revision.ppt
AaronChi5
 
PPTX
1.2 algebraic expressions t
math260
 
PPTX
11.4
nglaze10
 
PPTX
11.2
nglaze10
 
PPTX
6 addition and subtraction ii x
Tzenma
 
PPTX
16 partial fraction decompositions x
math266
 
PPTX
1.3 solving equations t
math260
 
PPT
2/27/12 Special Factoring - Sum & Difference of Two Cubes
jennoga08
 
13 multiplication and division of rational expressions
elem-alg-sample
 
Chapter-5-Quadratic-expressions-and-equations.pdf
SueWei1
 
Expresiones algebraicas pdf
MineidaSuarez
 
Factoring polynomials
Mark Ryder
 
Algebra Expressions and Exponents Math Presentation in Pink Purple Graphic St...
EdlynMalangPanti
 
Factoring2
MartinGeraldine
 
P6 factoring
salamhello
 
P6 factoring
salamhello
 
Polynomials
Mark Ryder
 
Em01 ba
nahomyitbarek
 
Factorization of algebraic expressions math
deepanoel
 
1.2 algebraic expressions t
math260
 
Algebra Revision.ppt
AaronChi5
 
1.2 algebraic expressions t
math260
 
11.4
nglaze10
 
11.2
nglaze10
 
6 addition and subtraction ii x
Tzenma
 
16 partial fraction decompositions x
math266
 
1.3 solving equations t
math260
 
2/27/12 Special Factoring - Sum & Difference of Two Cubes
jennoga08
 

More from alg1testreview (20)

PPTX
56 system of linear equations
alg1testreview
 
PPTX
55 addition and subtraction of rational expressions
alg1testreview
 
PPTX
52 rational expressions
alg1testreview
 
PPTX
51 basic shapes and formulas
alg1testreview
 
PPTX
41 expressions
alg1testreview
 
PPTX
59 constructing linea equations of lines
alg1testreview
 
PPTX
57 graphing lines from linear equations
alg1testreview
 
PPTX
58 slopes of lines
alg1testreview
 
PPTX
55 inequalities and comparative statements
alg1testreview
 
PPTX
56 the rectangular coordinate system
alg1testreview
 
PPTX
54 the number line
alg1testreview
 
PPTX
53 pythagorean theorem and square roots
alg1testreview
 
PPTX
52 about triangles
alg1testreview
 
PPTX
50 solving equations by factoring
alg1testreview
 
PPTX
51 ratio-proportion
alg1testreview
 
PPTX
49 factoring trinomials the ac method and making lists
alg1testreview
 
PPTX
48 factoring out the gcf and the grouping method
alg1testreview
 
PPTX
47 operations of 2nd degree expressions and formulas
alg1testreview
 
PPTX
45scientific notation
alg1testreview
 
PPTX
42 linear equations
alg1testreview
 
56 system of linear equations
alg1testreview
 
55 addition and subtraction of rational expressions
alg1testreview
 
52 rational expressions
alg1testreview
 
51 basic shapes and formulas
alg1testreview
 
41 expressions
alg1testreview
 
59 constructing linea equations of lines
alg1testreview
 
57 graphing lines from linear equations
alg1testreview
 
58 slopes of lines
alg1testreview
 
55 inequalities and comparative statements
alg1testreview
 
56 the rectangular coordinate system
alg1testreview
 
54 the number line
alg1testreview
 
53 pythagorean theorem and square roots
alg1testreview
 
52 about triangles
alg1testreview
 
50 solving equations by factoring
alg1testreview
 
51 ratio-proportion
alg1testreview
 
49 factoring trinomials the ac method and making lists
alg1testreview
 
48 factoring out the gcf and the grouping method
alg1testreview
 
47 operations of 2nd degree expressions and formulas
alg1testreview
 
45scientific notation
alg1testreview
 
42 linear equations
alg1testreview
 

Recently uploaded (20)

PPTX
202450812 BayCHI UCSC-SV 20250812 v17.pptx
home
 
PDF
LDMMIA Reiki Yoga Finals Review Spring Summer
LDMMia Reiki Lectures
 
PDF
Hazard Identification & Risk Assessment .pdf
estagiarioprotegesms
 
PDF
ChatGPT for Dummies - Pam Baker Ccesa007.pdf
Demetrio Ccesa Rayme
 
PDF
AI-driven educational solutions for real-life interventions in the Philippine...
EleniIlkou
 
PDF
OBE - B.A.(HON'S) IN INTERIOR ARCHITECTURE -Ar.MOHIUDDIN.pdf
ArchitectAFMMohiuddi
 
PDF
My India Quiz Book_20210205121199924.pdf
AtandraDey2
 
PPTX
A powerpoint presentation on the Revised K-10 Science Shaping Paper
JericEstaco2
 
PDF
Computing-Curriculum for Schools in Ghana
abigailanane203
 
PDF
Indian roads congress 037 - 2012 Flexible pavement
offersjackpot
 
PDF
medical_surgical_nursing_10th_edition_ignatavicius_TEST_BANK_pdf.pdf
victor kamau
 
PDF
IGGE1 Understanding the Self1234567891011
rhea22labucay
 
PDF
Chinmaya Tiranga quiz Grand Finale.pdf
Nitin Suresh
 
PDF
Trump Administration's workforce development strategy
Mebane Rash
 
PDF
David L Page_DCI Research Study Journey_how Methodology can inform one's prac...
David L Page
 
DOC
Soft-furnishing-By-Architect-A.F.M.Mohiuddin-Akhand.doc
ArchitectAFMMohiuddi
 
PDF
A GUIDE TO GENETICS FOR UNDERGRADUATE MEDICAL STUDENTS
DrBincy A. S
 
PDF
FORM 1 BIOLOGY MIND MAPS and their schemes
arcade5
 
PPTX
ELIAS-SEZIURE AND EPilepsy semmioan session.pptx
eyoba2172
 
PPTX
Unit 4 Computer Architecture Multicore Processor.pptx
Gunasundari Selvaraj
 
202450812 BayCHI UCSC-SV 20250812 v17.pptx
home
 
LDMMIA Reiki Yoga Finals Review Spring Summer
LDMMia Reiki Lectures
 
Hazard Identification & Risk Assessment .pdf
estagiarioprotegesms
 
ChatGPT for Dummies - Pam Baker Ccesa007.pdf
Demetrio Ccesa Rayme
 
AI-driven educational solutions for real-life interventions in the Philippine...
EleniIlkou
 
OBE - B.A.(HON'S) IN INTERIOR ARCHITECTURE -Ar.MOHIUDDIN.pdf
ArchitectAFMMohiuddi
 
My India Quiz Book_20210205121199924.pdf
AtandraDey2
 
A powerpoint presentation on the Revised K-10 Science Shaping Paper
JericEstaco2
 
Computing-Curriculum for Schools in Ghana
abigailanane203
 
Indian roads congress 037 - 2012 Flexible pavement
offersjackpot
 
medical_surgical_nursing_10th_edition_ignatavicius_TEST_BANK_pdf.pdf
victor kamau
 
IGGE1 Understanding the Self1234567891011
rhea22labucay
 
Chinmaya Tiranga quiz Grand Finale.pdf
Nitin Suresh
 
Trump Administration's workforce development strategy
Mebane Rash
 
David L Page_DCI Research Study Journey_how Methodology can inform one's prac...
David L Page
 
Soft-furnishing-By-Architect-A.F.M.Mohiuddin-Akhand.doc
ArchitectAFMMohiuddi
 
A GUIDE TO GENETICS FOR UNDERGRADUATE MEDICAL STUDENTS
DrBincy A. S
 
FORM 1 BIOLOGY MIND MAPS and their schemes
arcade5
 
ELIAS-SEZIURE AND EPilepsy semmioan session.pptx
eyoba2172
 
Unit 4 Computer Architecture Multicore Processor.pptx
Gunasundari Selvaraj
 

53 multiplication and division of rational expressions

  • 1. Multiplication and Division of Rational Expressions Frank Ma © 2011
  • 2. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions
  • 3. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel.
  • 4. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3
  • 5. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z
  • 6. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz
  • 7. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) *
  • 8. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 )
  • 9. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x + 2 )(x – 2)
  • 10. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) (x + 2 )(x – 2)
  • 11. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2)
  • 12. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2)
  • 13. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2)
  • 14. Multiplication Rule for Rational Expressions A B C D * = AC BD Multiplication and Division of Rational Expressions In most problems, we reduce the product by factoring the top and the bottom, then cancel. Example A. Simplify 10x y3z a. * y2 5x3 = 10xy2 5x3y3z = 2 x2yz b. (x2 + 2x – 3 ) (x – 2) (x2 – x ) (x2 – 4 ) = (x + 3)(x + 2) x * = (x + 3)(x – 1 ) (x – 2) x(x – 1 ) (x + 2 )(x – 2) In the next section, we meet the following type of problems.
  • 15. Multiplication and Division of Rational Expressions Example B. Simplify and expand the answers. a. x + 3 x – 1 (x2 – 1)
  • 16. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) Example B. Simplify and expand the answers.
  • 17. Multiplication and Division of Rational Expressions Example B. Simplify and expand the answers. a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1)
  • 18. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 Example B. Simplify and expand the answers.
  • 19. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) Example B. Simplify and expand the answers.
  • 20. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) Example B. Simplify and expand the answers.
  • 21. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) – x + 1 (x – 3)(x + 1) Example B. Simplify and expand the answers.
  • 22. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) Example B. Simplify and expand the answers.
  • 23. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) Example B. Simplify and expand the answers.
  • 24. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) Example B. Simplify and expand the answers.
  • 25. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) Example B. Simplify and expand the answers.
  • 26. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) = (x – 2)(x + 1) + (–x –1)(x + 3) Example B. Simplify and expand the answers.
  • 27. Multiplication and Division of Rational Expressions a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) = (x – 2)(x + 1) + (–x –1)(x + 3) = x2 – x – 2 – x2 – 4x – 3 Example B. Simplify and expand the answers.
  • 28. Multiplication and Division of Rational Expressions Example B. Simplify and expand the answers. a. x + 3 x – 1 (x2 – 1) = x + 3 (x – 1) (x – 1)(x + 1) = (x + 3)(x + 1) = x2 + 4x + 3 b. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) = x – 2 (x – 3)(x + 3) [ – x + 1 (x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3) = (x – 2)(x + 1) – (x + 1)(x + 3) = (x – 2)(x + 1) + (–x –1)(x + 3) = x2 – x – 2 – x2 – 4x – 3 = –5x – 5
  • 29. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷
  • 30. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate
  • 31. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate We convert division by an expression of multiplying by its reciprocal.
  • 32. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 33. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 34. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 35. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 36. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 37. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate Example C. Simplify (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) * (9 – x2) We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 38. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) * (–1) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 39. Division Rule for Rational Expressions Multiplication and Division of Rational Expressions A B C D ÷ = AD BC Reciprocate (2x – 6) (x + 3) ÷ (x2 + 2x – 3) (9 – x2) = (2x – 6) (x + 3) (x2 + 2x – 3) (9 – x2) * = 2(x – 3) (x + 3) (x + 3)(x – 1) (3 – x)(3 + x) * (–1) = –2(x – 1) (3 + x) Example C. Simplify We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
  • 40. Multiplication and Division of Rational Expressions Besides the expanded form and factored forms, rational expressions may also be split into sums or differences.
  • 41. Multiplication and Division of Rational Expressions Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this.
  • 42. Multiplication and Division of Rational Expressions Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
  • 43. Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (2x – 6) 3x2 b. =
  • 44. Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. =
  • 45. Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2
  • 46. Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2 = – 2 x2 2 3x
  • 47. Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2 = – 2 x2 2 3x II. Long Division Long division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable.
  • 48. Multiplication and Division of Rational Expressions Example D. Break up the numerators as the sums or differences and simplify each term. (2x – 6) (x + 3) a. = Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term. (x + 3) – (x + 3) 2x 6 (2x – 6) 3x2 b. = – 2x 6 3x2 3x2 = – 2 x2 2 3x II. Long Division Long division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable. Specifically, long division gives relevant results only when the degree of the numerator is the same or more than the degree of the denominator.
  • 49. Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division.
  • 50. Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient.
  • 51. Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
  • 52. Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 1
  • 53. Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 1 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45
  • 54. Multiplication and Division of Rational Expressions Let’s look at the example 125/8 or 125 ÷ 8 by long division. i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 1 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered.
  • 55. Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 15 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered. 40 5 Let’s look at the example 125/8 or 125 ÷ 8 by long division.
  • 56. Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 15 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: N D = Q + R D and that R (the remainder) is smaller then D (no more quotient). 40 5 where Q is the quotient Let’s look at the example 125/8 or 125 ÷ 8 by long division.
  • 57. Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125 15 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 8 45 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: N D = Q + R D and that R (the remainder) is smaller then D (no more quotient). 40 5 125 8 = 15 + 5 8 where Q is the quotient Let’s look at the example 125/8 or 125 ÷ 8 by long division.
  • 58. Multiplication and Division of Rational Expressions Example E. Divide using long division(2x – 6) (x + 3)
  • 59. Multiplication and Division of Rational Expressions i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 60. Multiplication and Division of Rational Expressions )x + 3 2x – 6 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Make sure the terms are in order. Example E. Divide using long division(2x – 6) (x + 3)
  • 61. Multiplication and Division of Rational Expressions )x + 3 2x – 6 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Make sure the terms are in order. Example E. Divide using long division(2x – 6) (x + 3)
  • 62. Multiplication and Division of Rational Expressions )x + 3 2x – 6 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. enter the quotients of the leading terms 2x/x = 2 Example E. Divide using long division(2x – 6) (x + 3)
  • 63. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. enter the quotients of the leading terms 2x/x = 2 Example E. Divide using long division(2x – 6) (x + 3)
  • 64. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 65. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 2x + 6 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 66. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 –) i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 67. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. –) i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 68. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 –) Stop. No more quotient since x can’t going into 12. iii. Repeat steps i and ii until no more quotient may be entered. i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 69. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 70. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) Hence we may write (2x – 6) (x + 3) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example E. Divide using long division(2x – 6) (x + 3)
  • 71. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) Hence we may write (2x – 6) (x + 3) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Q R Example E. Divide using long division(2x – 6) (x + 3)
  • 72. Multiplication and Division of Rational Expressions )x + 3 2x – 6 2 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. 2x + 6 –12 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: = 2 – 12 x + 3 –) Hence we may write (2x – 6) (x + 3) N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Q R Q R Example E. Divide using long division(2x – 6) (x + 3)
  • 73. Multiplication and Division of Rational Expressions Example F. Divide using long division ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 x – 2 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
  • 74. Multiplication and Division of Rational Expressions )x + 3 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 Make sure the terms are in order. N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 75. Multiplication and Division of Rational Expressions )x + 3 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 76. Multiplication and Division of Rational Expressions )x + 3 x ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 77. Multiplication and Division of Rational Expressions )x + 3 x ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 78. Multiplication and Division of Rational Expressions )x + 3 x ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 79. Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 80. Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 81. Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27–) 30 N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 82. Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient –) x2 – 6x + 3 –9x – 27–) 30 Stop. No more quotient since x can’t going into 30. Hence 30 is the remainder. and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 83. Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27–) 30 Hence x2 – 6x + 3 x – 2 = N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 84. Multiplication and Division of Rational Expressions )x + 3 x – 9 ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator. x2 + 3x –9x + 3 iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form: –) x2 – 6x + 3 –9x – 27–) 30 Hence x2 – 6x + 3 x – 2 = x – 9 + 30 x + 3 i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms. N D = Q + R D has smaller degree then denominator D (no more quotient). where Q is the quotient and the remainder R Example F. Divide using long divisionx2 – 6x + 3 x – 2
  • 85. Ex A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions 1. 10x * 2 5x3 15x 4 * 16 25x4 10x * 35x32. 5. 10 9x4 * 18 5x3 6. 3.12x6* 5 6x14 56x6 27 * 63 8x5 10x * 35x34. 7. 75x 49 * 42 25x3 8. 9. 2x – 4 2x + 4 5x + 10 3x – 6 10. 6 – 4x 3x – 2 x – 2 2x + 4 11. 9x – 12 2x – 4 2 – x 8 – 6x 12. x + 4 –x – 4 4 – x x – 4 13. 3x – 9 15x – 5 3 – x 5 – 15x 14. 42 – 6x –2x + 14 4 – 2x –7x + 14 * * * * * * 15. (x2 + x – 2 ) (x – 2) (x2 – x) (x2 – 4 ) * 16. (x2 + 2x – 3 ) (x2 – 9) (x2 – x – 2 ) (x2 – 2x – 3) * 17. (x2 – x – 2 ) (x2 – 1) (x2 + 2x + 1) (x2 + x ) * 18. (x2 + 5x – 6 ) (x2 + 5x + 6) (x2 – 5x – 6 ) (x2 – 5x + 6) * 19. (x2 – 3x – 4 ) (x2 – 1) (x2 – 2x – 8) (x2 – 3x + 2) * 20. (– x2 + 6 – x ) (x2 + 5x + 6) (x2 – x – 12 ) (6 – x2 – x) *
  • 86. Ex. A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions 21. (2x2 + x – 1 ) (1 – 2x) (4x2 – 1) (2x2 – x ) 22. (3x2 – 2x – 1) (1 – 9x2) (x2 + x – 2 ) (x2 + 4x + 4) 23.(3x2 – x – 2) (x2 – x + 2) (3x2 + 4x + 1) (–x – 3x2) 24. (x + 1 – 6x2) (–x2 – 4) (2x2 + x – 1 ) (x2 – 5x – 6) 25. (x3 – 4x) (–x2 + 4x – 4) (x2 + 2) (–x + 2) 26. (–x3 + 9x ) (x2 + 6x + 9) (x2 + 3x) (–3x2 – 9x) Ex. B. Multiply, expand and simplify the results. ÷ ÷ ÷ ÷ ÷ ÷ 27. x + 3 x + 1 (x2 – 1) 28. x – 3 x – 2 (x2 – 4) 29. 2x + 3 1 – x (x2 – 1) 30. 3 – 2x x + 2 (x + 2)(x +1) 31. 3 – 2x 2x – 1 (3x + 2)(1 – 2x) 32. x – 2 x – 3 ( + x + 1 x + 3 )( x – 3)(x + 3) 33. 2x – 1 x + 2 ( – x + 2 2x – 3 ) ( 2x – 3)(x + 2)
  • 87. Multiplication and Division of Rational Expressions 38. x – 2 x2 – 9 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1) 39. x + 3 x2 – 4 ( – 2x + 1 x2 + x – 2 ) ( x – 2)(x + 2)(x – 1) 40. x – 1 x2 – x – 6 ( – x + 1 x2 – 2x – 3 ) ( x – 3)(x + 2)(x + 1) 41. x + 2 x2 – 4x +3 ( – 2x + 1 x2 + 2x – 3 )( x – 3)(x + 3)(x – 1) 34. 4 – x x – 3 ( – x – 1 2x + 3 )( x – 3)(2x + 3) 35. 3 – x x + 2 ( – 2x + 3 x – 3 )(x – 3)(x + 2) Ex B. Multiply, expand and simplify the results. 36. 3 – 4x x + 1 ( – 1 – 2x x + 3 )( x + 3)(x + 1) 37. 5x – 7 x + 5 ( – 4 – 5x x – 3 )(x – 3)(x + 5)
  • 88. Ex. C. Break up the following expressions as sums and differences of fractions. 42. 43. 44. 45. 46. 47. x2 + 4x – 6 2x2x2 – 4 x2 12x3 – 9x2 + 6x 3x x2 – 4 2x x x8 – x6 – x4 x2 x8 – x6 – x4 Ex D. Use long division and write each rational expression in the form of Q + . R D (x2 + x – 2 ) (x – 1) (3x2 – 3x – 2 ) (x + 2) 2x + 6 x + 2 48. 3x – 5 x – 2 49. 4x + 3 x – 1 50. 5x – 4 x – 3 51. 3x + 8 2 – x 52. –4x – 5 1 – x 53. 54. (2x2 + x – 3 ) (x – 2) 55. 56. (–x2 + 4x – 3 ) (x – 3) (5x2 – 1 ) (x – 4) 57. (4x2 + 2 ) (x + 3) 58. 59. Multiplication and Division of Rational Expressions