# Notes 8.3 - The Remainder and Factor Theorems

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10 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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Notes 8.3
-

The Remainder and Factor Theorems

The Remainder Theorem
:

Here are three ways to find a remainder…

For the polynomial: f(x) = x
4

-

3
x
3

+
x
2

-

3x + 1, divide it by
x
-

2
.

Long Division:

Synthetic Division:

Finding f(x):

(you may

just want to watch this)

f(2
) =
2
4

3(2)
3

+ 2
2

3(2) + 1

x
3

-

x
2

-

x

-

5

2 1
-
3 1
-
3 1 16

3(8) + 4
-

6 + 1

2
-
2

-
2
-
10

16

24 + 4

6 + 1

-

(
x
4

-

2x
3
)

1

-
1
-
1

-
5
-
9

-
8 + 4

6 + 1

-
4

6 + 1

-
x
3

+ x
2

-
10 + 1

-
(
-
x
3

+ 2x
2
)

-
9

-
x
2

-

3x

-
(
-
x
2

+ 2x)

-
5x + 1

Look how lucky

you are that

-
(
-
5x + 10)

someone discovered this!

-
9

What would be the remainder if x

8 was a factor of

x
4

+ 2x
3

10x
2

+ 5x

7
?

ZERO!

Quotient

If a polynomial f(x) is divided by x

a, the remainder is the
constant f(a).

Coefficients of the
Quotient

Remainder

Remainder

The Factor Theorem

The binomial x

a is a fa
ctor of the
polynomial f(x) if and only if f(a) = 0.

Is x

4 a factor of the polynomial
f(x) = x
4

+ x
3

13x
2

25x

12
?

f(4) = 4
4

+ 4
3

13(4)
2

25(4)

12

OR

4 1 1
-
13
-
25
-
12

= 256 + 64

208

100

12

4 20 28 12

= 0

YES

1 5 7 3 0

Ex 1) Use synthetic division to find f(4) for

f(x) = x
4

6x
3

+ 8x
2

+ 5x + 13

(Put the divisor in the box, list the coefficient
s in order,

bring down the first term, multiply each answer by the divisor, then add down)

4

1

-
6

8

5

13

+

4

-
8

0

20

1

-
2

0

5

33

f(4) = 33

Ex 2) Use synthetic divisi
on to determine if x+2 is a factor of the function,

f(x) = 3x
5

5x
3

+ 57

-
2

3

0

-
5

0

0

57

+

-
6

12

-
14

28

-
56

3

-
6

7

-
14

28 1

f(
-
2) = 1, so x+2 is not a factor.

KEY

The remainder

Remainder

Divide using synthetic division. Is the binomial a factor of the
polynomial?

Ex 3)
(x
3

64)

(x

4)

4

1

0

0

-
64

+ 4

16 64

1

4

16

0

yes

Ex 4)
(x
4

x
3

+ 2x

2)
(x

2)

2

1

-
1

0

2

-
2

+ 2 2 4 12

1

1

2

6

10

no

KEY

Ex 5) Show that x

2

is a factor of
x
3

+ 7x
2

+ 2x

40
.

Then find the remaining factors.

2

1

7

2

-
40

+ 2 18 40

1

9

20

0

(x

2)(x
2

+ 9x + 20)

(x

2)(x + 5)(x + 4)

so, x
3

+ 7x
2

+ 2x

40 =

(x

2)(x + 5)(x + 4)

Ex 6) Show that 2x + 7 is a factor of
2x
3

+ 17x
2

+ 23x

42.

Then find the remainin
g
factors.

2

17

2

3
-
42

+
-
7
-
35 42

2

10

-
12

0

(Since you used 2
as the denominator, you must divide 2 out of
the resulting expression)

2

+ 10x

12 it is x
2

+ 5x

6

(2x + 7)(x
2

+ 5x

6)

(2x + 7)(x

1)(x + 6)

so, 2x
3

+ 17x
2

+ 23x

42 = (2x + 7)(x

1)(x + 6)

KEY