The document discusses dividing polynomials using long division and synthetic division. It provides examples of dividing polynomials using long division, including inserting placeholders for missing terms. It then introduces synthetic division as a shortcut method for dividing polynomials when the divisor is of the form x - k. Examples are provided of using synthetic division to divide polynomials.

1. Dividing Polynomials
Chapter 5 Polynomial and Rational Functions

2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Use long division to divide polynomials.
⚫ Use synthetic division to divide polynomials.

3. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −

4. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −
2m‒1  4m3 ‒ 8m2 + 4m + 6

5. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −
2m2
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 – 2m2
You have to cancel
out the first term,
so this has to be
the same.

6. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −
2m2
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
–
Now change all of
the signs.

7. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −
2m2 ‒ 3m
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
–6m2 + 3m
–

8. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −
2m2 ‒ 3m
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
+6m2 – 3m
m + 6
–

9. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −
2m2 ‒ 3m + ½
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
+6m2 – 3m
m + 6
m – ½
–

10. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1
m m m m
− + + −
2m2 ‒ 3m + ½
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
+6m2 – 3m
m + 6
m + ½
13
2
–
–
12
6
2
 
=
 
 

11. Dividing Polynomials
⚫ Therefore,
⚫ If either polynomial does not have a term for each
power, you will need to insert a placeholder for it.
3 2
2
13
4 8 4 6 1 2
2 3
2 1 2 2 1
m m m
m m
m m
− + +
= − + +
− −

12. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150

13. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
3x
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150
3x3 + 0x2 ‒ 12x
‒ 2x2 + 12x ‒ 150

14. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
3x ‒ 2
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150
3x3 + 0x2 ‒ 12x
‒ 2x2 + 12x ‒ 150
‒ 2x2 + 0x + 8
12x ‒ 158

15. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
So,
3x ‒ 2
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150
3x3 + 0x2 ‒ 12x
‒ 2x2 + 12x ‒ 150
‒ 2x2 + 0x + 8
12x ‒ 158
3 2
2 2
3 2 150 12 158
3 2
4 4
x x x
x
x x
− − −
= − +
− −

16. Dividing Polynomials
⚫ More formally, we can state that:
Let f(x) and g(x) be polynomials with g(x) of degree one
or more, but of lower degree than f(x). There exist
unique polynomials q(x) and r(x) such that
where either r(x) = 0 or the degree of r(x) is less than the
degree of g(x). ( ) ( ) ( ) ( )
= +
f x g x q x r x

17. Dividing Polynomials (cont.)
⚫ For example, could be evaluated as
or
− −
−
3 2
2
3 2 150
4
x x
x
− − + −
2 3 2
3
4 3 2 0 150
x
x x x x
− + +
3 2
3 0 12
x x x
− + −
2
2 12 150
x x
−2
+ −
2
2 0 8
x x
−
12 158
x
−
− +
−
2
12 158
3 2
4
x
x
x

18. Dividing Polynomials (cont.)
⚫ Using the division algorithm, this means that
( )( )
− − = − − + −
3 2 2
3 2 150 4 3 2 12 158
x x x x x
( )
f x ( )
g x ( )
q x ( )
r x
Dividend = Divisor • Quotient + Remainder

19. Synthetic Division
⚫ A shortcut method of performing long division with
certain polynomials, called synthetic division, is used
only when a polynomial is divided by a binomial of the
form x – k, where the coefficient of x is 1.
⚫ To use synthetic division:
The numbers on the bottom are the coefficients of the
quotient.
−1 1 0
...
n n
k a a a a
−
−
+ + + +
=
−
1
1 1 0
...
n n
n n
a x a x a x a
x k
an
kan
− +
1
n n
a ka …

20. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + −
−
3 2
4 15 11 10
3
x x x
x

21. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + −
−
3 2
4 15 11 10
3
x x x
x
− −
3 4 15 11 10

22. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + −
−
3 2
4 15 11 10
3
x x x
x
− −
3 4 15 11 10
4
12
–3

23. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + −
−
3 2
4 15 11 10
3
x x x
x
− −
3 4 15 11 10
4
12
–3
–9
2

24. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + −
−
3 2
4 15 11 10
3
x x x
x
− −
3 4 15 11 10
4
12
–3
–9
2
6
–4

25. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + −
−
3 2
4 15 11 10
3
x x x
x
− −
3 4 15 11 10
4
12
–3
–9
2
6
–4
−
− + +
−
2 4
4 3 2
3
x x
x
Notice that the
exponent is 1 less
than the numerator.

26. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + − +
−
4 3
3 15 50 25
4
x x x
x

27. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + − +
−
4 3
3 15 50 25
4
x x x
x
− −
4 3 15 50
0 25
Notice the place-
holder for the
missing x2 term.

28. Synthetic Division (cont.)
⚫ Example: Use synthetic division to divide
− + − +
−
4 3
3 15 50 25
4
x x x
x
− −
4 3 15 50
0 25
−3
–12
3
12
12
48
−2
–8
17
− + + − +
−
3 2 17
3 3 12 2
4
x x x
x
Notice the place-
holder for the
missing x2 term.

29. Synthetic Division (cont.)
If the coefficient of x is not 1, we can divide everything in
both expressions by the coefficient to still let us use
synthetic division.
⚫ Example: Use synthetic division to divide
+ − −
−
3 2
2 7 13 3
2 3
x x x
x

30. Synthetic Division (cont.)
If the coefficient of x is not 1, we can divide everything in
both expressions by the coefficient to still let us use
synthetic division.
⚫ Example: Use synthetic division to divide
First, we have to divide everything by 2 (the
coefficient of 2x).
+ − −
−
3 2
2 7 13 3
2 3
x x x
x

31. Synthetic Division (cont.)
Now, we can set up the synthetic division:
+ − −
+ − −
 =
− −
3 2
3 2 1
2
1
2
13 3
7
2 7 13 3 2 2 2
3
2 3
2
x x x
x x x
x x
− −
3 13 3
7
1
2 2 2 2
1 5 1 0
3
2
15
2
3
2
2
5 1
x x
+ +

32. Classwork
⚫ College Algebra 2e
⚫ 5.4: 16-36 (×4); 5.3: 30-46 (even); 5.2: 52-64 (even)
⚫ 5.4 Classwork Check
⚫ Quiz 5.3