q8

Q: Find the value of k in the following polynomials when

(i) (x+3) is a factor of (5x+4)³ - (3x+k)³

(ii) (x-2) is a factor of (3x+7)³ - (4x+k)³

Solution:

(i)

Let P(x) = (5x+4)³ - (3x+k)³

(x+3) is a factor of P(x)

Using converse of factor theorem, we can say that P(-3)=0

P(-3)=[5(-3)+4]³ - [3(-3)+k]³

Therefore, [5(-3)+4]³ - [3(-3)+k]³ = 0

[-15+4]³ - [-9+k]³ = 0

(-11)³ - (-9+k)³ = 0

(-9+k)³ = 11³

-9+k = 11

k = 20

 

(ii)

Let Q(x)=(3x+7)³ - (4x+k)³

(x-2) is a factor of Q(x)

Using converse of factor theorem, we can say that Q(2)=0

Q(2)=[3(2)+7]³ - [4(2)+k]³

[3(2)+7]³ - [4(2)+k]³ = 0

[6+7]³ - [8+k]³ = 0

(13)³ - (8+k)³ = 0

(8+k)³ = 13³

8+k = 13

k = 5