1. Warm Up Divide using long division. 1. 12.18 ÷ 2.1 5.8 Divide.
6x – 15y 3 2.
2x + 5y
7a2 – ab a 3.
7a – b

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

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. 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

6. 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

7. 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).

9. 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.

10. 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 –

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

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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

18. Check It Out! Example 3
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.

19. Lesson Quiz
1. Divide by using long division. (8x3 + 6x2 + 7) ÷ (x + 2)
8x2 – 10x + 20 – 33
x + 2
2. Divide by using synthetic division. (x3 – 3x + 5) ÷ (x + 2)
x2 – 2x + 1 + 3
x + 2
3. Find an expression for the height of a parallelogram whose area is represented by 2x3 – x2 – 20x + 3 and whose base is represented by (x + 3).
2x2 – 7x + 1