This document provides instructions on dividing polynomials using long division and synthetic division. It explains that long division can be used to divide polynomials just as it is used to divide numbers. An example of long division of polynomials is shown. Synthetic division is introduced as an alternative method that is used when the coefficient of the highest term is 1. Examples of both long division and synthetic division of polynomials are provided and step-by-step workings are shown. Some notes are included that synthetic division should be used when the coefficient is 1 and that testing for zero remainder determines if a binomial is a factor.

1. Dividing Polynomials

2. Warm Up
• Without a calculator,
divide the following
113277323
Solution: 49251

3. This long division technique can also be
used to divide polynomials

4. POLYNOMIALS – DIVIDING
EX – Long division
• (5x³ -13x² +10x -8) / (x-2)
5x³ - 13x² + 10x - 8x - 2
5x²
5x³ - 10x²-( )
-3x² + 10x
- 3x
-3x² + 6x- ( )
4x - 8
4x - 8- ( )
+ 4
0
R 0

5. So in other words…
5x³ -13x² +10x - 8
x-2
= 5x² -3x + 4
(5x² -3x + 4)(x-2) = 5x³ -13x² +10x - 8
OR

6. POLYNOMIALS – DIVIDING
EX#2 – Long division
• (x² +3x -12) / (x+6)
x² + 3x - 12x - 3
x
x² - 3x- ( )
6x -12
+ 6
6x - 18- ( )
6
R 6

7. • (2x² -19x + 8) / (x-8)
2x² - 19x + 8x - 8
Let’s Try One

8. • (2x² -19x + 8) / (x-8)
2x² - 19x + 8x - 8
Let’s Try One

9. EX – Synthetic division
(5x³ -13x² +10x -8) / (x-2)
2 5 -13 10 -8
5
10
-3
-6
4
8
0
5x² -3x + 4
Opposite
of number in
divisor

10. EX – Synthetic division
(3x³ -4x² +2x -1) / (x+1)
-1 3 -4 2 -1
3
-3
-7
+7
9
-9
-10
3x2 -7x + 9 R-10
Opposite
of number in
divisor

11. Let’s Try One
(x³ -13x +12) / (x+4)

12. EX – Synthetic division
(x³ -13x +12) / (x+4)Opposite
of number in
divisor

13. A Couple of Notes
• Use synthetic division when the coefficient in
front of x is 1
(x- 2) (2x-3)122
 xx 122
 xx
YES NO
• To test so see if a binomial is a factor, you want
to see if you get a remainder of zero. If yes, it is a
factor. If you get a remainder, the answer is no.

14. • (2x² -19x + 8) / (x-8)
2x² - 19x + 8x - 8
From this example, x-8 IS a factor
because the remainder is zero

15. In this case, x-3 is not a factor because
there was a remainder of 6
x² + 3x - 12x - 3
x
x² - 3x- ( )
6x -12
+ 6
6x - 18- ( )
6
R 6

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