Question

verify whether f(x)=4x^3+4x^2-x-1 is exactly divisible by 2x+1​

Answer 1

Given that,

f(x) = 4x³ + 4x² - x - 1

g(x) = 2x + 1

We have to verify whether f(x) is exactly divisible by g(x)

First, Let us find the factors of g(x)

⇒ g(x) = 2x + 1

⇒ x = -1/2

The factor of g(x) is -1/2, So if g(x) completely divides f(x) then f(-1/2) must be equal to zero.

⇒ f(-1/2)

⇒ 4(-1/2)³ + 4(-1/2)² - (-1/2) - 1

⇒ 4×-1/8 + 4×1/4 + 1/2 - 1

⇒ -1/2 + 1 + 1/2 - 1

⇒ 0

Since, the remainder is zero:

∴ f(x) is exactly divisible by 2x + 1

Some Information :-

☛ A polynomial f(x) is divisible by the factors of another polynomial p(x) if and only if polynomial p(x) completely divides the polynomial f(x), This is also called Factor theorem.

Answer 2

Step-by-step explanation:

4x^3+4x^2-x-1

4x^2(x+1)-1(x+1)

=(x+1).(4x^2-1)

=(x-1) {(2x)^2-(1)^2}

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

(2x+1) is a factor,so it is exactly divisible by (2x+1)