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Notes - Polynomial Division

L
Lori Rapp

1. Write the dividend and divisor with the divisor outside the long division bar. 2. Divide the first term of the dividend by the divisor and write the result above the division bar. 3. Multiply the divisor by the result and write the product below the terms of the dividend. 4. Subtract to find the remainder and bring down the next term to continue the process until there is no remainder. This document explains how to perform long division of polynomials using the same process as long division of numbers. It provides an example of performing long division step-by-step to divide a quadratic polynomial by a linear polynomial.

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Polynomial Division Dividing a Monomial by a Monomial Long Division Synthetic Division
Dividing a Monomial by a Monomial Apply the rules for dividing exponents:
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: Like bases, subtract smaller from the larger exponent to get the new exponent and keep the same base.
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: Like bases, subtract smaller from the larger exponent to get the new exponent and keep the same base. If the larger exponent is in the numerator, the result goes in the numerator.
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. If the larger exponent is in the numerator, the result goes in the numerator.
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the numerator.
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the 2 numerator. =x
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the 2 numerator. =x If the larger exponent is in the denominator, the result goes in the denominator.
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 a4 Like bases, subtract smaller from 3 9 the larger exponent to get the x a new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the 2 numerator. =x If the larger exponent is in the denominator, the result goes in the denominator.
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 a 4 Like bases, subtract smaller from 3 9 the larger exponent to get the x a new exponent and keep the same base. =x 5− 3 1 = 9− 4 If the larger exponent is in the a numerator, the result goes in the 2 numerator. =x If the larger exponent is in the denominator, the result goes in the denominator.
Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 a 4 Like bases, subtract smaller from 3 9 the larger exponent to get the x a new exponent and keep the same base. =x 5− 3 1 = 9− 4 If the larger exponent is in the a numerator, the result goes in the 2 numerator. =x 1 If the larger exponent is in the = 5 denominator, the result goes in a the denominator.
Simplify. 5 3 8a b 2 7 6a b
Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 6 by 2.
Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b
Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b Apply the rules for dividing powers with like bases.
Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b Apply the rules for 4a 5−2 dividing powers 7− 3 with like bases. 3b
Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b Apply the rules for 4a 5−2 dividing powers 7− 3 with like bases. 3b 3 And you are done 4a 4 dividing a monomial 3b by a monomial.
Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk
Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 2 −1 12 f = 10 − 8 5 30 d
Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 2 −1 12 f = 10 − 8 5 30 d 2f = 2 5d
Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 3 12 f 2 −1 27 j = 10 − 8 = 9− 3 9−5 5 30 d 1 9h jk 2f = 2 5d
Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 3 12 f 2 −1 27 j = 10 − 8 = 9− 3 9−5 5 30 d 1 9h jk 2f 3 = 2 = 6 4 5d h k
Algebra Cruncher Problems Follow this link to try a couple on your own at Cool Math. Notice when you select the “Give me a Problem” button to try new problems, 2 rows are generated. Look carefully between them. That red line indicates this problem is a fraction. Do your work in a notebook before entering your answer. When you select “What’s the Answer?” compare your answer with the given answer. Keep selecting new problems until you get 3 consecutive problems correct.
Divide a Polynomial by a Monomial Visit this Cool math website to learn about dividing a Polynomial by a monomial. Be sure to click the “next page” to review the 2 pages of notes. Complete the “Try it” problem on page 2 in your notebook.
Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2
Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2 2 2 4wy 20w 6wy 7 = 2 − 2 + 2 − 2 4wy 4wy 4wy 4wy
Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2 2 2 4wy 20w 6wy 7 = 2 − 2 + 2 − 2 4wy 4wy 4wy 4wy 2 5 2 −1 3 4wy 20 w 6w 7 = − 2 + 2 −1 − 2 4wy 2 1 4y 2 4 wy 4wy
Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2 2 2 4wy 20w 6wy 7 = 2 − 2 + 2 − 2 4wy 4wy 4wy 4wy 2 5 2 −1 3 4wy 20 w 6w 7 = − 2 + 2 −1 − 2 4wy 2 1 4y 2 4 wy 4wy 5w 3 7 = 1− 2 + − 2 y 2y 4wy
It’s Practice time... Go to the Regents Prep website to practice dividing a polynomial by a monomial. Only practice questions 1 through 7! Message or Pronto me if you have questions.
Polynomial Long Division Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper polynomial as the sum of a polynomial with a remainder.
Polynomial Long Division Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper polynomial as the sum of a polynomial with a remainder. You can also use polynomial division to help you factor polynomials completely. Just as we can divide 56 by 7, we can divide x2 + 4x – 32 by x + 8.
Polynomial Long Division Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper polynomial as the sum of a polynomial with a remainder. You can also use polynomial division to help you factor polynomials completely. Just as we can divide 56 by 7, we can divide x2 + 4x – 32 by x + 8. Explore Long Division here by viewing this video.
Let’s try one together! ( x + 3x − 12 ) divided by ( x − 3) 2 2 The expression x + 3x – 12 is called the dividend and the expression x -3 is called the divisor.
Let’s try one together! ( x + 3x − 12 ) divided by ( x − 3) 2 2 The expression x + 3x – 12 is called the dividend and the expression x -3 is called the divisor. Divisor goes outside the division bar.
Let’s try one together! ( x + 3x − 12 ) divided by ( x − 3) 2 2 The expression x + 3x – 12 is called the dividend and the expression x -3 is called the divisor. Divisor goes outside the division bar. 2 x−3 x + 3x − 12
2 x−3 x + 3x − 12
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x 2 x−3 x + 3x − 12
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x 2 x−3 x + 3x − 12
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs!
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! ( 2 − x − 3x )
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! ( 2 − x − 3x ) 6x
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x?
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x?
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it under the portion left over from before.
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder.
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x + 6 R6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder.
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x + 6 R6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder. Write your final answer as a sum with the remainder written over the divisor.
Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x + 6 R6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder. 6 x+6+ Write your final answer as a sum with x−3 the remainder written over the divisor.
Try this one on your own... ( x + 3x − 6x − 7 ) ÷ ( x + 4 ) 3 2 Clip art licensed from the Clip Art Gallery on DiscoverySchool.com
2 1 Final answer: x − x − 2 + x+4 2 x −x −2 R1 3 2 x+4 x +3x −6x −7 3 2 − (x +4x ) 2 −x −6x 2 − (−x −4x) −2x −7 − (−2x −8) 1
Need more work with long division? Follow this link to the MathIsFun website to see more examples of long divison
Synthetic Division We also have a simplified method called Synthetic Division that takes away the variables and the messy subtraction! Watch this video on Synthetic Division and be amazed!
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3)
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve.
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. x−3= 0 x=3
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 x=3
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 x=3 3
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 3
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 3 1 3 −12
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 1
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the first coefficient and place the product under the next coefficient. 1
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 first coefficient and place the product under the next coefficient. 1
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 first coefficient and place the product under the next coefficient. 1 6. Add down and repeat the process.
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 first coefficient and place the product under the next coefficient. 1 6 6. Add down and repeat the process.
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6. Add down and repeat the process.
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6 6. Add down and repeat the process.
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6 6. Add down and repeat the process. 7. Write your polynomial starting with 1 less degree than your divisor. The last digit will be your remainder.
See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6 6. Add down and repeat the process. 6 7. Write your polynomial starting with 1 less x+6+ degree than your divisor. The last digit x−3 will be your remainder.
You try! x + 3x - 6x – 7 3 2 divided by x+4
You try! Find the number for the box. x + 3x - 6x – 7 3 2 divided by x+4
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 divided by x+4
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 divided by −4 x+4
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 x+4
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 x+4
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 x+4 1
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 x+4 1 −1
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 4 x+4 1 −1 −2
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 4 8 x+4 1 −1 −2 1
You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 4 8 x+4 1 −1 −2 1 2 1 Final Answer: x − x − 2 + x+4
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division.
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 1
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 1
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 1 −4
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 1 −4
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 1 −4 2
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2
( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2 0
Try this one... ( 3 x − 14x + 8 ÷ x + 4) ( ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2 0 The remainder is 0 so it’s not included in the result.
Try this one... ( 3 x − 14x + 8 ÷ x + 4 ) ( ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2 0 2 x − 4x + 2 The remainder is 0 so it’s not included in the result.
Shall we Play a GAME? Check your knowledge on Dividing Polynomials by playing Jeopardy. Ok, technically it’s called Challenge Board but it’s the same idea! There are 4 categories: horseshoes, handgrenades, doesn’t count, and polynomial long division. You have the option to play alone or against a friend or family member. You could even arrange a time with a classmate to meet on Pronto to play. Try the App Share feature to see the same game board!
Remainder Theorem If the polynomial P(x) of degree n ≥ 1 is divided by (x –a), where a is a constant, then the remainder is P(a).
Remainder Theorem If the polynomial P(x) of degree n ≥ 1 is divided by (x –a), where a is a constant, then the remainder is P(a). In English, this means we can use the remainder theorem to evaluate a function for a specific value. See it in action…
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 3
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 3
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 3 −11
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 3 −11
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 3 −11 29
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29 −68
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29 Therefore, f(-2) is −68
Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29 Therefore, f(-2) is −68 Don’t believe it? Check it out in your calculator.
Congratulations! You’ve finished the notes and practice for Polynomial Division. You are now ready to proceed to the Assignment. Good luck!

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Notes - Polynomial Division

  • 1. Polynomial Division Dividing a Monomial by a Monomial Long Division Synthetic Division
  • 2. Dividing a Monomial by a Monomial Apply the rules for dividing exponents:
  • 3. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: Like bases, subtract smaller from the larger exponent to get the new exponent and keep the same base.
  • 4. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: Like bases, subtract smaller from the larger exponent to get the new exponent and keep the same base. If the larger exponent is in the numerator, the result goes in the numerator.
  • 5. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. If the larger exponent is in the numerator, the result goes in the numerator.
  • 6. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the numerator.
  • 7. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the 2 numerator. =x
  • 8. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 Like bases, subtract smaller from 3 the larger exponent to get the x new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the 2 numerator. =x If the larger exponent is in the denominator, the result goes in the denominator.
  • 9. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 a4 Like bases, subtract smaller from 3 9 the larger exponent to get the x a new exponent and keep the same base. =x 5− 3 If the larger exponent is in the numerator, the result goes in the 2 numerator. =x If the larger exponent is in the denominator, the result goes in the denominator.
  • 10. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 a 4 Like bases, subtract smaller from 3 9 the larger exponent to get the x a new exponent and keep the same base. =x 5− 3 1 = 9− 4 If the larger exponent is in the a numerator, the result goes in the 2 numerator. =x If the larger exponent is in the denominator, the result goes in the denominator.
  • 11. Dividing a Monomial by a Monomial Apply the rules for dividing exponents: x 5 a 4 Like bases, subtract smaller from 3 9 the larger exponent to get the x a new exponent and keep the same base. =x 5− 3 1 = 9− 4 If the larger exponent is in the a numerator, the result goes in the 2 numerator. =x 1 If the larger exponent is in the = 5 denominator, the result goes in a the denominator.
  • 12. Simplify. 5 3 8a b 2 7 6a b
  • 13. Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 6 by 2.
  • 14. Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b
  • 15. Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b Apply the rules for dividing powers with like bases.
  • 16. Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b Apply the rules for 4a 5−2 dividing powers 7− 3 with like bases. 3b
  • 17. Simplify. 5 8a b 3 Reduce the 2 7 numerical part by 6a b dividing the 8 and 4 8a b 5 3 6 by 2. 2 7 3 6a b Apply the rules for 4a 5−2 dividing powers 7− 3 with like bases. 3b 3 And you are done 4a 4 dividing a monomial 3b by a monomial.
  • 18. Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk
  • 19. Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 2 −1 12 f = 10 − 8 5 30 d
  • 20. Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 2 −1 12 f = 10 − 8 5 30 d 2f = 2 5d
  • 21. Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 3 12 f 2 −1 27 j = 10 − 8 = 9− 3 9−5 5 30 d 1 9h jk 2f = 2 5d
  • 22. Simplify each of the following. 8 2 3 5 12d f 27h jk 10 9 9 30d f 9h jk 2 3 12 f 2 −1 27 j = 10 − 8 = 9− 3 9−5 5 30 d 1 9h jk 2f 3 = 2 = 6 4 5d h k
  • 23. Algebra Cruncher Problems Follow this link to try a couple on your own at Cool Math. Notice when you select the “Give me a Problem” button to try new problems, 2 rows are generated. Look carefully between them. That red line indicates this problem is a fraction. Do your work in a notebook before entering your answer. When you select “What’s the Answer?” compare your answer with the given answer. Keep selecting new problems until you get 3 consecutive problems correct.
  • 24. Divide a Polynomial by a Monomial Visit this Cool math website to learn about dividing a Polynomial by a monomial. Be sure to click the “next page” to review the 2 pages of notes. Complete the “Try it” problem on page 2 in your notebook.
  • 25. Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2
  • 26. Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2 2 2 4wy 20w 6wy 7 = 2 − 2 + 2 − 2 4wy 4wy 4wy 4wy
  • 27. Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2 2 2 4wy 20w 6wy 7 = 2 − 2 + 2 − 2 4wy 4wy 4wy 4wy 2 5 2 −1 3 4wy 20 w 6w 7 = − 2 + 2 −1 − 2 4wy 2 1 4y 2 4 wy 4wy
  • 28. Try It - Page 2 ( 4wy − 20w + 6wy − 7 ) ÷ 4wy 2 2 2 2 2 4wy 20w 6wy 7 = 2 − 2 + 2 − 2 4wy 4wy 4wy 4wy 2 5 2 −1 3 4wy 20 w 6w 7 = − 2 + 2 −1 − 2 4wy 2 1 4y 2 4 wy 4wy 5w 3 7 = 1− 2 + − 2 y 2y 4wy
  • 29. It’s Practice time... Go to the Regents Prep website to practice dividing a polynomial by a monomial. Only practice questions 1 through 7! Message or Pronto me if you have questions.
  • 30. Polynomial Long Division Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper polynomial as the sum of a polynomial with a remainder.
  • 31. Polynomial Long Division Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper polynomial as the sum of a polynomial with a remainder. You can also use polynomial division to help you factor polynomials completely. Just as we can divide 56 by 7, we can divide x2 + 4x – 32 by x + 8.
  • 32. Polynomial Long Division Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper polynomial as the sum of a polynomial with a remainder. You can also use polynomial division to help you factor polynomials completely. Just as we can divide 56 by 7, we can divide x2 + 4x – 32 by x + 8. Explore Long Division here by viewing this video.
  • 33. Let’s try one together! ( x + 3x − 12 ) divided by ( x − 3) 2 2 The expression x + 3x – 12 is called the dividend and the expression x -3 is called the divisor.
  • 34. Let’s try one together! ( x + 3x − 12 ) divided by ( x − 3) 2 2 The expression x + 3x – 12 is called the dividend and the expression x -3 is called the divisor. Divisor goes outside the division bar.
  • 35. Let’s try one together! ( x + 3x − 12 ) divided by ( x − 3) 2 2 The expression x + 3x – 12 is called the dividend and the expression x -3 is called the divisor. Divisor goes outside the division bar. 2 x−3 x + 3x − 12
  • 36. 2 x−3 x + 3x − 12
  • 37. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x 2 x−3 x + 3x − 12
  • 38. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x 2 x−3 x + 3x − 12
  • 39. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs!
  • 40. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! ( 2 − x − 3x )
  • 41. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! ( 2 − x − 3x ) 6x
  • 42. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x
  • 43. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12
  • 44. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x?
  • 45. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x?
  • 46. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it under the portion left over from before.
  • 47. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6
  • 48. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x +6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder.
  • 49. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x + 6 R6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder.
  • 50. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x + 6 R6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder. Write your final answer as a sum with the remainder written over the divisor.
  • 51. Start by doing monomial by monomial division. Divide the 1st term of the dividend an divisor. x2/x = x x + 6 R6 Write the product under the terms in the dividend. Then subtract the 2 product from the portion above it. x −3 x + 3x − 12 Watch your signs! Next, bring down the next term in the ( 2 − x − 3x ) dividend. 6x −12 Repeat the same steps. What is 6x/x? Multiply the divisor again and write it − ( 6x − 18 ) under the portion left over from before. 6 When you are out of terms to bring down, the last number is the remainder. 6 x+6+ Write your final answer as a sum with x−3 the remainder written over the divisor.
  • 52. Try this one on your own... ( x + 3x − 6x − 7 ) ÷ ( x + 4 ) 3 2 Clip art licensed from the Clip Art Gallery on DiscoverySchool.com
  • 53. 2 1 Final answer: x − x − 2 + x+4 2 x −x −2 R1 3 2 x+4 x +3x −6x −7 3 2 − (x +4x ) 2 −x −6x 2 − (−x −4x) −2x −7 − (−2x −8) 1
  • 54. Need more work with long division? Follow this link to the MathIsFun website to see more examples of long divison
  • 55. Synthetic Division We also have a simplified method called Synthetic Division that takes away the variables and the messy subtraction! Watch this video on Synthetic Division and be amazed!
  • 56. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3)
  • 57. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve.
  • 58. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. x−3= 0 x=3
  • 59. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 x=3
  • 60. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 x=3 3
  • 61. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 3
  • 62. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 3 1 3 −12
  • 63. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12
  • 64. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 1
  • 65. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the first coefficient and place the product under the next coefficient. 1
  • 66. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 first coefficient and place the product under the next coefficient. 1
  • 67. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 first coefficient and place the product under the next coefficient. 1 6. Add down and repeat the process.
  • 68. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 first coefficient and place the product under the next coefficient. 1 6 6. Add down and repeat the process.
  • 69. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6. Add down and repeat the process.
  • 70. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6 6. Add down and repeat the process.
  • 71. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6 6. Add down and repeat the process. 7. Write your polynomial starting with 1 less degree than your divisor. The last digit will be your remainder.
  • 72. See our long division example using synthetic division! (x2 + 3x – 12) divided by (x – 3) 1. Set the divisor equal to zero and solve. 2. Put that number in a box in the top left corner. x−3= 0 3. List the coefficients of the dividend in x=3 order from left to right. 4. Drop the first coefficient. 3 1 3 −12 5. Multiply the number in the box by the 3 18 first coefficient and place the product under the next coefficient. 1 6 6 6. Add down and repeat the process. 6 7. Write your polynomial starting with 1 less x+6+ degree than your divisor. The last digit x−3 will be your remainder.
  • 73. You try! x + 3x - 6x – 7 3 2 divided by x+4
  • 74. You try! Find the number for the box. x + 3x - 6x – 7 3 2 divided by x+4
  • 75. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 divided by x+4
  • 76. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 divided by −4 x+4
  • 77. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 x+4
  • 78. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 x+4
  • 79. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 x+4 1
  • 80. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 x+4 1 −1
  • 81. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 4 x+4 1 −1 −2
  • 82. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 4 8 x+4 1 −1 −2 1
  • 83. You try! Find the number for the box. x+4=0 x + 3x - 6x – 7 3 2 x = −4 Coefficients. divided by −4 1 3 −6 −7 −4 4 8 x+4 1 −1 −2 1 2 1 Final Answer: x − x − 2 + x+4
  • 84. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division.
  • 85. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8
  • 86. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 1
  • 87. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 1
  • 88. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 1 −4
  • 89. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 1 −4
  • 90. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 1 −4 2
  • 91. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2
  • 92. ( Try this one... 3 ) ( x − 14x + 8 ÷ x + 4 ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2 0
  • 93. Try this one... ( 3 x − 14x + 8 ÷ x + 4) ( ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2 0 The remainder is 0 so it’s not included in the result.
  • 94. Try this one... ( 3 x − 14x + 8 ÷ x + 4 ) ( ) Notice the x2 term is missing. You need to account for this term when setting up your division. −4 1 0 −14 8 −4 16 −8 1 −4 2 0 2 x − 4x + 2 The remainder is 0 so it’s not included in the result.
  • 95. Shall we Play a GAME? Check your knowledge on Dividing Polynomials by playing Jeopardy. Ok, technically it’s called Challenge Board but it’s the same idea! There are 4 categories: horseshoes, handgrenades, doesn’t count, and polynomial long division. You have the option to play alone or against a friend or family member. You could even arrange a time with a classmate to meet on Pronto to play. Try the App Share feature to see the same game board!
  • 96. Remainder Theorem If the polynomial P(x) of degree n ≥ 1 is divided by (x –a), where a is a constant, then the remainder is P(a).
  • 97. Remainder Theorem If the polynomial P(x) of degree n ≥ 1 is divided by (x –a), where a is a constant, then the remainder is P(a). In English, this means we can use the remainder theorem to evaluate a function for a specific value. See it in action…
  • 98. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2
  • 99. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10
  • 100. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 3
  • 101. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 3
  • 102. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 3 −11
  • 103. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 3 −11
  • 104. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 3 −11 29
  • 105. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29
  • 106. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29 −68
  • 107. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29 Therefore, f(-2) is −68
  • 108. Evaluate the function using the remainder theorem. f ( x ) = 3x − 5x + 7x − 10 at x = −2 3 2 −2 3 −5 7 −10 −6 22 −58 3 −11 29 Therefore, f(-2) is −68 Don’t believe it? Check it out in your calculator.
  • 109. Congratulations! You’ve finished the notes and practice for Polynomial Division. You are now ready to proceed to the Assignment. Good luck!

Editor's Notes