At the end of the lesson, students should

be able to:

use the remainder and factor theorems.

LEARNING OUTCOMES

Consider

The remainder may be

obtained by long division

as follows

2267 23 xxxx45

10

84

24

105

65

2

2672

2

2

2

23

23

xx

x

x

xx

xx

xx

xxxx

10245267 223 xxxxxx

Remainder

Alternatively,

2P

10

02x 2xLet

2267 23 xxxx

22627223

267 23 xxxxPLet

DEFINITIONWhen a polynomial P(x)

is divided by a linear

factor (x-a), then the

remainder is P(a)

Remainder

Theorem

Proof:

P(x) = (x-a) Q(x) + R(x)

When x= a;

P(a) = 0 + R(x)

= R(x)

P(x) = D(x) Q(x) + R(x)

EXAMPLE

By using remainder theorem, find the

remainder when is

divided by

62 23 xxxP

3x

By remainder theorem

Remainder

SOLUTION

3P

632323

39

03x 3xLet

YOUR TURN!!

Using remainder theorem, find the remainder for

the following

i.

ii.

iii.

iv.

2 ; 137 23 xxxxxP

xxxxP ; 122

2 ; 232 2 xxxxP

1 ; 534

xxxP

When a polynomial P(x) = x3 + px2 + qx -9

is divided by x+1, the remainder is 3 and

when P(x) is divided by x-2, the remainder is

9. Find the value of p and q.

EXAMPLE

When is divided by

(x + 2), the remainder is 16. Determine k.

865 234 xxkxx

EXAMPLE

Let

Given that

865)( 234 xxkxxxP

16)2(P

168)2(6)2(5)2()2( 234 k

248k

3k

SOLUTION

a) Given that .

When P(x) is divided by , the

remainder is twice of the remainder

when P(x) is divided by . Find a .

EXERCISE

162)( 23 xaxxxP

2x

1x

EXAMPLE

By using remainder theorem, find the

remainder when is

divided by

276 2 xxxP

12x

If the remainder obtained from dividing the

polynomial P(x) by (x-a) is zero, then the linear term

(x-a) is called a factor of the polynomial P(x).

If P(a) = 0 then (x a) is a factor of P(x)