Question

Find the value of a and b so that the polynomial x3

– 4x2 + ax + b is exactly divisible by x – 2 as well

as x + 1.​

Answer 1

Question :

Find the value of a and b so that the polynomial

x^3 - 4x^2 +ax + b is exactly divisible by x-2 as well as

x + 1.

Solution :

Dividing x^3 - 4x^2 + ax + b by x-2

________________

x -2 ) x^3 - 4x^2 + ax + b ( x^2 -2x + (a-4)

x^3 - 2x^2

- +

_________________

-2x^2 + ax + b

-2x^2 + 4x

+ -

______________

x (a-4) + b

x (a-4) -2(a-4)

- +

_______________

b +2a- 8

It is given that

x -2 will exactly divide x^3-4x^2 +ax+b

so, remainder will be zero

b + 2a - 8= 0.....eqn(1)

Dividing x^3 -4x^2+ax +b by x+1

_______________

x +1 ) x^3 -4x^2 +ax + b ( x^2-5x + (a+5)

x^3 +x^2

- -

______________

-5x^2 +ax+b

-5x^2 -5x

+ +

___________

x (a+5) +b

x (a+5) +(a+5)

- -

__________

b - a -5

again

b - a - 5 = 0

b = a + 5 .... eqn(2)

putting value of b in eqn(1)

(a+5) +2a-8 = 0

3 a -3 = 0

a = 1

putting value of a In eqn(2)

b = 1 + 5

b = 6

Hence,

the value of

a is 1

and

b is 6.