Question

If a polynomial x
4
-3x
3
-6x
2+kx-16 is exactly divisible by x
2
-3x+2 then find the value of

k

Answer 1

Answer:

The value of k is 24.

Step-by-step explanation:

Let,

f(x)=x^4-3x^3-6x^2+kx -16f(x)=x

4

−3x

3

−6x

2

+kx−16

Given,

f(x) is exactly divisible by x^2-3x+2x

2

−3x+2

That is,

f(x) is a multiple of x^2-3x+2x

2

−3x+2

Also,

x^2-3x+2=x^2-2x-x+2=x(x-2)-1(x-2)=(x-1)(x-2)x

2

−3x+2=x

2

−2x−x+2=x(x−2)−1(x−2)=(x−1)(x−2)

⇒ f(x) is a multiple of (x-1)(x-2)(x−1)(x−2)

⇒ f(x) is a multiple of (x-1),

⇒ x = 1 is one of the zeroes of f(x)

⇒ f(1) = 0

⇒ (1)^4-3(1)^3-6(1)^2+k(1) -16=0(1)

4

−3(1)

3

−6(1)

2

+k(1)−16=0

⇒ 1-3-6+k -16=0\implies k-24=0\implies k = 241−3−6+k−16=0⟹k−24=0⟹k=24

Hence, The value of k is 24.