1. Objective
Use long division and synthetic division to divide polynomials.

2. Vocabulary
synthetic division

3. Polynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.

4. Example 1: Using Long Division to Divide a Polynomial
Divide using long division.
(–y2 + 2y3 + 25) ÷ (y – 3)
Step 1 Write the dividend in standard form, including
terms with a coefficient of 0.
2y3 – y2 + 0y + 25
Step 2 Write division in the same way you would when dividing numbers.
y – 3 2y3 – y2 + 0y + 25

5. Example 1 Continued
Step 3 Divide.
2y2
+ 5y
+ 15
Notice that y times 2y2 is 2y3. Write 2y2 above 2y3.
y – 3 2y3 – y2 + 0y + 25
–(2y3 – 6y2)
Multiply y – 3 by 2y2. Then subtract. Bring down the next term. Divide 5y2 by y.
5y2 + 0y
–(5y2 – 15y)
Multiply y – 3 by 5y. Then subtract. Bring down the next term. Divide 15y by y.
15y + 25
Multiply y – 3 by 15. Then subtract.
–(15y – 45) 70
Find the remainder.

6. Example 1 Continued
Step 4 Write the final answer.
–y2 + 2y3 + 25
y – 3
= 2y2 + 5y 70
y – 3

7. Check It Out! Example 1a
Divide using long division.
(15x2 + 8x – 12) ÷ (3x + 1)
Step 1 Write the dividend in standard form, including
terms with a coefficient of 0.
15x2 + 8x – 12
Step 2 Write division in the same way you would when dividing numbers.
3x x2 + 8x – 12

8. Check It Out! Example 1a Continued
Step 3 Divide. 5x
+ 1
Notice that 3x times 5x is 15x2. Write 5x above 15x2.
3x x2 + 8x – 12
–(15x2 + 5x)
Multiply 3x + 1 by 5x. Then subtract. Bring down the next term. Divide 3x by 3x.
3x – 12
–(3x + 1)
Multiply 3x + 1 by 1. Then subtract.
–13
Find the remainder.

9. Check It Out! Example 1a Continued
Step 4 Write the final answer.
15x2 + 8x – 12
3x + 1
= 5x + 1 – 13
3x + 1

10. Check It Out! Example 1b
Divide using long division.
(x2 + 5x – 28) ÷ (x – 3)
Step 1 Write the dividend in standard form, including
terms with a coefficient of 0.
x2 + 5x – 28
Step 2 Write division in the same way you would when dividing numbers.
x – 3 x2 + 5x – 28

11. Check It Out! Example 1b Continued
Step 3 Divide. x
+ 8
Notice that x times x is x2. Write x above x2.
x – 3 x2 + 5x – 28
–(x2 – 3x)
Multiply x – 3 by x. Then subtract. Bring down the next term. Divide 8x by x.
8x – 28
–(8x – 24)
Multiply x – 3 by 8. Then subtract. –4
Find the remainder.

12. Check It Out! Example 1b Continued
Step 4 Write the final answer.
x2 + 5x – 28
x – 3
= x + 8 – 4
x – 3

13. Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form (x – a).

15. Example 2A: Using Synthetic Division to Divide by a Linear Binomial
Divide using synthetic division. 1 3
(3x2 + 9x – 2) ÷ (x – )
Step 1 Find a. Then write the coefficients and a in the synthetic division format. 1 3
a =
For (x – ), a = . 1 3 1 3 –2
Write the coefficients of 3x2 + 9x – 2.

16. Example 2A Continued
Step 2 Bring down the first coefficient. Then multiply and add for each column. 1 3 –2 1 3
Draw a box around the remainder, 1 3 1 1 3 3 10
Step 3 Write the quotient. 3x 1 3
x –

17. Example 2A Continued 3x 1 3
x –
Check Multiply (x – )
(x – ) 1 3 3x
x –
= 3x2 + 9x – 2

18. Example 2B: Using Synthetic Division to Divide by a Linear Binomial
Divide using synthetic division.
(3x4 – x3 + 5x – 1) ÷ (x + 2)
Step 1 Find a.
a = –2
For (x + 2), a = –2.
Step 2 Write the coefficients and a in the synthetic division format.
3 – –1 –2
Use 0 for the coefficient of x2.

19. Example 2B Continued
Step 3 Bring down the first coefficient. Then multiply and add for each column. –2
3 – –1
Draw a box around the remainder, 45. –6 14
–28 46 3 –7 14
–23 45
Step 4 Write the quotient.
3x3 – 7x2 + 14x – 23 + 45
x + 2
Write the remainder over the divisor.

20. Check It Out! Example 2a
Divide using synthetic division.
(6x2 – 5x – 6) ÷ (x + 3)
Step 1 Find a.
a = –3
For (x + 3), a = –3.
Step 2 Write the coefficients and a in the synthetic division format. –3
6 –5 –6
Write the coefficients of 6x2 – 5x – 6.

21. Check It Out! Example 2a Continued
Step 3 Bring down the first coefficient. Then multiply and add for each column. –3
6 –5 –6
Draw a box around the remainder, 63.
–18 69 6
–23 63
Step 4 Write the quotient.
6x – 23 + 63
x + 3
Write the remainder over the divisor.

22. Check It Out! Example 2b
Divide using synthetic division.
(x2 – 3x – 18) ÷ (x – 6)
Step 1 Find a.
a = 6
For (x – 6), a = 6.
Step 2 Write the coefficients and a in the synthetic division format. 6
1 –3 –18
Write the coefficients of x2 – 3x – 18.

23. Check It Out! Example 2b Continued
Step 3 Bring down the first coefficient. Then multiply and add for each column. 6
1 –3 –18
There is no remainder. 6 18 1 3
Step 4 Write the quotient.
x + 3

24. You can use synthetic division to evaluate polynomials
You can use synthetic division to evaluate polynomials. This process is called synthetic substitution. The process of synthetic substitution is exactly the same as the process of synthetic division, but the final answer is interpreted differently, as described by the Remainder Theorem.

25. Example 3A: Using Synthetic Substitution
Use synthetic substitution to evaluate the polynomial for the given value.
P(x) = 2x3 + 5x2 – x + 7 for x = 2. 2
–1 7
Write the coefficients of the dividend. Use a = 2. 4 18 34 2 9 17 41
P(2) = 41
Check Substitute 2 for x in P(x) = 2x3 + 5x2 – x + 7.
P(2) = 2(2)3 + 5(2)2 – (2) + 7
P(2) = 41 

26. Example 3B: Using Synthetic Substitution
Use synthetic substitution to evaluate the polynomial for the given value. 1 3
P(x) = 6x4 – 25x3 – 3x + 5 for x = – . – 1 3
6 – –3 5
Write the coefficients of the dividend. Use 0 for the coefficient of x2. Use a = . 1 3 –2 9 –3 2 6
–27 9 –6 7
P( ) = 7 1 3

27. Check It Out! Example 3a
Use synthetic substitution to evaluate the polynomial for the given value.
P(x) = x3 + 3x2 + 4 for x = –3. –3
Write the coefficients of the dividend. Use 0 for the coefficient of x2 Use a = –3. –3 1 4
P(–3) = 4
Check Substitute –3 for x in P(x) = x3 + 3x2 + 4.
P(–3) = (–3)3 + 3(–3)2 + 4
P(–3) = 4 

28. Check It Out! Example 3b
Use synthetic substitution to evaluate the polynomial for the given value. 1 5
P(x) = 5x2 + 9x + 3 for x = . 1 5
Write the coefficients of the dividend. Use a = . 1 5 1 2 5 10 5
P( ) = 5 1 5

29. Example 4: Geometry Application
Write an expression that represents the area of the top face of a rectangular prism when the height is x + 2 and the volume of the prism is x3 – x2 – 6x.
The volume V is related to the area A and the
height h by the equation V = A  h. Rearranging for A gives A = . V h
x3 – x2 – 6x
x + 2
A(x) =
Substitute. –2
1 –1 –6 0
Use synthetic division.
The area of the face of the rectangular prism can be represented by A(x)= x2 – 3x. –2 6 1 –3

30. Check It Out! Example 4
Write an expression for the length of a rectangle with width y – 9 and area y2 – 14y + 45.
The area A is related to the width w and the length l by the equation A = l  w.
y2 – 14y + 45
y – 9
l(x) =
Substitute. 9
1 –14 45
Use synthetic division. 9
–45 1 –5
The length of the rectangle can be represented by l(x)= y – 5.