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The document discusses operations that can be performed on polynomials. It defines terms and like-terms, and explains that like-terms can be combined. It provides examples of expanding and simplifying polynomials by distributing terms, multiplying terms, and combining like-terms. Key operations covered include distributing numbers to polynomial terms, multiplying terms by multiplying their coefficients and variables, and multiplying two polynomials by multiplying each term of one against the other.

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Polynomial Operations
Polynomial Operations In the last section, we introduced polynomial expressions. #xN ± #xN-1 ± … ± #x ± #
Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Hence, 3(5x) = (3*5)x =15x, Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. When multiplying a number to a polynomial, we may expand the result using the distributive law: A(B ± C) = AB ± AC. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
Example A. Expand and simplify. Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable.
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = b. 3x2(–4x) = c. 3x2(2x3 – 4x) =
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 b. 3x2(–4x) = c. 3x2(2x3 – 4x) =
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 = 6x5 b. 3x2(–4x) = c. 3x2(2x3 – 4x) =
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 = 6x5 b. 3x2(–4x) = 3(–4)x2x = –12x3 c. 3x2(2x3 – 4x) =
Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 = 6x5 b. 3x2(–4x) = 3(–4)x2x = –12x3 c. 3x2(2x3 – 4x) distribute = 6x5 – 12x3
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. = 3x(2x – 1) + 2(2x – 1) a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1) Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1) Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP.
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1) Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP. A shorter way to multiply is to bypass the 2nd step and use the general distributive law.
General Distributive Rule: Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac .. Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc .. Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x + 6
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x + 6 = x3– 5x2 + 4x + 6 We will address the division operation of polynomials later- after we understand more about the multiplication operation.
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x + 6 = x3– 5x2 + 4x + 6 We will address the division operation of polynomials later- after we understand more about the multiplication operation.
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 x3
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, Polynomial Operations (x – 3)(x2 – 2x – 2) = x3
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 –3x2
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 –3x2 –2x2
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2 –3x2 –2x2
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Next collect the x–terms Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x –2x 6x
We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x. The last term is the number term or –3(–2) = 6. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x + 6 (same as before) 6
Ex. A. Multiply the following monomials. 1. 3x2(–3x2) 11. 4x(3x – 5) – 9(6x – 7) Polynomial Operations 2. –3x2(8x5) 3. –5x2(–3x3) 4. –12( ) 6 –5x3 5. 24( x3) 8 –5 6. 6x2( ) 3 2x3 7. –15x4( x5) 5 –2 C. Expand and simplify. B. Fill in the degrees of the products. 8. #x(#x2 + # x + #) = #x? + #x? + #x? 9. #x2(#x4 + # x3 + #x2) = #x? + #x? + #x? 10. #x4(#x3 + # x2 + #x + #) = #x? + #x? + #x? + #x? 12. –x(2x + 7) + 3(4x – 2) 13. –3x(3x + 2) – 8x(7x – 5) 14. 5x(–5x + 9) + 6x(6x – 1) 15. 2x(–4x + 2) – 3x(2x – 1) – 3(4x – 2) 16. –4x(–7x + 9) – 2x(2x – 5) + 9(4x + 2)
18. (x + 5)(x + 7) Polynomial Operations D. Expand and simplify. (Use any method.) 19. (x – 5)(x + 7) 20. (x + 5)(x – 7) 21. (x – 5)(x – 7) 22. (3x – 5)(2x + 4) 23. (–x + 5)(3x + 8) 24. (2x – 5)(2x + 5) 25. (3x + 7)(3x – 7) 26. (3x2 – 5)(x – 6) 27. (8x – 2)(–4x2 – 7) 28. (2x – 7)(x2 – 3x + 9) 29. (5x + 3)(2x2 – x + 5) 38. (x – 1)(x + 1) 39. (x – 1)(x2 + x + 1) 40. (x – 1)(x3 + x2 + x + 1) 41. (x – 1)(x4 + x3 + x2 + x + 1) 42. What do you think the answer is for (x – 1)(x50 + x49 + …+ x2 + x + 1)? 30. (x – 1)(x – 1) 31. (x + 1)2 32. (2x – 3)2 33. (5x + 4)2 34. 2x(2x – 1)(3x + 2) 35. 4x(3x – 2)(2x + 3) 36. (x – 5)(2x – 1)(3x + 2) 37. (2x + 1)(3x + 1)(x – 2)

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2 polynomial operations x

  • 2. Polynomial Operations In the last section, we introduced polynomial expressions. #xN ± #xN-1 ± … ± #x ± #
  • 3. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 4. Terms with the same variabe part are called like-terms. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 5. Terms with the same variabe part are called like-terms. Like-terms may be combined. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 6. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 7. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 8. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 9. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 10. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 11. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 12. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 13. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Hence, 3(5x) = (3*5)x =15x, Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 14. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 15. Terms with the same variabe part are called like-terms. Like-terms may be combined. For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2. Unlike terms may not be combined. In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it to the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. When multiplying a number to a polynomial, we may expand the result using the distributive law: A(B ± C) = AB ± AC. Polynomial Operations In the last section, we introduced polynomial expressions. Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms
  • 16. Example A. Expand and simplify. Polynomial Operations
  • 17. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) Polynomial Operations
  • 18. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 Polynomial Operations
  • 19. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x Polynomial Operations
  • 20. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 Polynomial Operations
  • 21. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) Polynomial Operations
  • 22. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 Polynomial Operations
  • 23. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 Polynomial Operations
  • 24. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations
  • 25. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable.
  • 26. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = b. 3x2(–4x) = c. 3x2(2x3 – 4x) =
  • 27. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 b. 3x2(–4x) = c. 3x2(2x3 – 4x) =
  • 28. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 = 6x5 b. 3x2(–4x) = c. 3x2(2x3 – 4x) =
  • 29. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 = 6x5 b. 3x2(–4x) = 3(–4)x2x = –12x3 c. 3x2(2x3 – 4x) =
  • 30. Example A. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Operations When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example B. a. (3x2)(2x3) = 3*2x2x3 = 6x5 b. 3x2(–4x) = 3(–4)x2x = –12x3 c. 3x2(2x3 – 4x) distribute = 6x5 – 12x3
  • 31. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations
  • 32. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. a. (3x + 2)(2x – 1)
  • 33. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. = 3x(2x – 1) + 2(2x – 1) a. (3x + 2)(2x – 1)
  • 34. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 a. (3x + 2)(2x – 1)
  • 35. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 a. (3x + 2)(2x – 1)
  • 36. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 a. (3x + 2)(2x – 1)
  • 37. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) a. (3x + 2)(2x – 1)
  • 38. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 a. (3x + 2)(2x – 1)
  • 39. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1)
  • 40. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1) Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.)
  • 41. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1) Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP.
  • 42. To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify. Polynomial Operations Example C. b. (2x – 1)(2x2 + 3x –4) = 3x(2x – 1) + 2(2x – 1) = 6x2 – 3x + 4x – 2 = 6x2 + x – 2 = 2x(2x2 + 3x –4) –1(2x2 + 3x – 4) = 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4 a. (3x + 2)(2x – 1) Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP. A shorter way to multiply is to bypass the 2nd step and use the general distributive law.
  • 44. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) Polynomial Operations
  • 45. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac .. Polynomial Operations
  • 46. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc .. Polynomial Operations
  • 47. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Polynomial Operations
  • 48. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) Polynomial Operations
  • 49. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 Polynomial Operations
  • 50. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x Polynomial Operations
  • 51. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x Polynomial Operations
  • 52. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 Polynomial Operations
  • 53. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 Polynomial Operations
  • 54. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations
  • 55. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3
  • 56. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2
  • 57. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x
  • 58. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2
  • 59. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x
  • 60. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x + 6
  • 61. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x + 6 = x3– 5x2 + 4x + 6 We will address the division operation of polynomials later- after we understand more about the multiplication operation.
  • 62. General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..) = Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc .. Example D. Expand a. (x + 3)(x – 4) = x2 – 4x + 3x – 12 simplify = x2 – x – 12 b. (x – 3)(x2 – 2x – 2) Polynomial Operations = x3 – 2x2 – 2x – 3x2 + 6x + 6 = x3– 5x2 + 4x + 6 We will address the division operation of polynomials later- after we understand more about the multiplication operation.
  • 63. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 x3
  • 64. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, Polynomial Operations (x – 3)(x2 – 2x – 2) = x3
  • 65. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 –3x2
  • 66. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 –3x2 –2x2
  • 67. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2 –3x2 –2x2
  • 68. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Next collect the x–terms Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2
  • 69. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x –2x 6x
  • 70. We include the following method for multiplying polynomials. Instead of expand all the terms then collect like–terms, we scan the like–terms and collect them starting from the highest degree term in the answer. We demonstrate this method and double check the answer in part b. Example E. Multiply a. (x – 3)(x2 – 2x – 2) The highest degree term in the product is the x3 term so we start with the x3–term. It’s x*x2 = 1x3. Next collect all the x2 terms in the product, they are –3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2. Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x. The last term is the number term or –3(–2) = 6. Polynomial Operations (x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x + 6 (same as before) 6
  • 71. Ex. A. Multiply the following monomials. 1. 3x2(–3x2) 11. 4x(3x – 5) – 9(6x – 7) Polynomial Operations 2. –3x2(8x5) 3. –5x2(–3x3) 4. –12( ) 6 –5x3 5. 24( x3) 8 –5 6. 6x2( ) 3 2x3 7. –15x4( x5) 5 –2 C. Expand and simplify. B. Fill in the degrees of the products. 8. #x(#x2 + # x + #) = #x? + #x? + #x? 9. #x2(#x4 + # x3 + #x2) = #x? + #x? + #x? 10. #x4(#x3 + # x2 + #x + #) = #x? + #x? + #x? + #x? 12. –x(2x + 7) + 3(4x – 2) 13. –3x(3x + 2) – 8x(7x – 5) 14. 5x(–5x + 9) + 6x(6x – 1) 15. 2x(–4x + 2) – 3x(2x – 1) – 3(4x – 2) 16. –4x(–7x + 9) – 2x(2x – 5) + 9(4x + 2)
  • 72. 18. (x + 5)(x + 7) Polynomial Operations D. Expand and simplify. (Use any method.) 19. (x – 5)(x + 7) 20. (x + 5)(x – 7) 21. (x – 5)(x – 7) 22. (3x – 5)(2x + 4) 23. (–x + 5)(3x + 8) 24. (2x – 5)(2x + 5) 25. (3x + 7)(3x – 7) 26. (3x2 – 5)(x – 6) 27. (8x – 2)(–4x2 – 7) 28. (2x – 7)(x2 – 3x + 9) 29. (5x + 3)(2x2 – x + 5) 38. (x – 1)(x + 1) 39. (x – 1)(x2 + x + 1) 40. (x – 1)(x3 + x2 + x + 1) 41. (x – 1)(x4 + x3 + x2 + x + 1) 42. What do you think the answer is for (x – 1)(x50 + x49 + …+ x2 + x + 1)? 30. (x – 1)(x – 1) 31. (x + 1)2 32. (2x – 3)2 33. (5x + 4)2 34. 2x(2x – 1)(3x + 2) 35. 4x(3x – 2)(2x + 3) 36. (x – 5)(2x – 1)(3x + 2) 37. (2x + 1)(3x + 1)(x – 2)