# FINDING REMAINDER USING REMAINDER THEOREM

Finding Remainder Using Remainder Theorem :

In this section, yo will learn how to find remainder using remainder theorem.

## Finding Remainder Using Remainder Theorem - Questions

Question 1 :

What is the remainder when

x2018 + 2018

is divided by (x – 1).

Solution :

Let

p(x)  =  x2018 + 2018

Here, the divisor is (x - 1).

Equate the divisor to zero.

x - 1  =  0

Solve for x.

x  =  1

Substitute 1 for x in p(x).

p(1)  =  (1)2018 + 2018

p(1)  =  1 + 2018

p(1)  =  2019

So, the remainder is 2019.

Question 2 :

For what value of k is the polynomial

p(x)  =  2x3 - kx2 + 3x + 10

exactly divisible by (x – 2).

Solution :

Here, the divisor is (x - 2).

Equate the divisor to zero.

x - 2  =  0

Solve for x.

x  =  2

Substitute 2 for x in p(2).

p(2)  =  2(2)3 - k(2)2 + 3(2) + 10

p(2)  =  2(8) - k(4) + 6 + 10

p(2)  =  16 - 4k + 16

p(2)  =  32 - 4k

So, the remainder is (32 - 4k).

If p(x) is exactly divisible by (x - 2), then the remainder must be zero.

Then,

32 - 4k  =  0

Solve for k.

32 - 4k  =  0

-4k  =  -32

k  =  8

Therefore, p(x) is exactly divisible by (x-2) when k  =  8.

Question 3 :

If two polynomials

2x3 + ax2 + 4x – 12 and x3 + x2 –2x + a

leave the same remainder when divided by (x – 3), find the value of a and also find the remainder.

Solution :

Here, the divisor is (x - 3).

Equate the divisor to zero.

x - 3  =  0

Solve for x.

x  =  3

Let

p(x)  =  2x3 + ax2 + 4x – 12 and q(x)  =  x3 + x2 –2x+ a

p(x)  =  2x3 + ax2 + 4x – 12

p(3)  =  2(3)3 + a(3)2 + 4(3) – 12

=  2(27) + a(9) + 12 - 12

p(3)  =  54 + 9a -------(1)

q(x)  =  x3 + x2 –2x+ a

q(3)  =  33 + 32 –2(3) + a

27 + 9 - 6 + a

30 + a  ------(2)

(1)  =  (2) ---->

54 + 9a  =  30 + a

9a - a  =  30 - 54

8a  =  -24

a  =  -3

Question 4 :

Determine whether (x -1) is a factor of the following polynomials:

(i) x3 + 5x2 - 10x + 4

Solution :

x - 1  =  0

x  =  1

p(x)  =  x3 + 5x2 - 10x + 4

p(1)  =  13 + 5(1)2 - 10(1) + 4

=  1 + 5 - 10 + 4

=  10 - 10

=  0

(ii)  x4 + 5x2 - 5x + 1

Solution :

x - 1  =  0

x  =  1

p(x)  =   x4 + 5x2 - 5x + 1

p(1)  =   14 + 5(1)2 - 5(1) + 1

=  1 + 5 - 5 + 1

=  2 After having gone through the stuff given above, we hope that the students would have understood how to find remainder using remainder theorem.

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