Question

If x ^ 3 + 6x ^ 2 + 4x + k is exactly divisible by (x + 2) , then k is equal to ​

Answer 1

`k=-8`

Explanation:

According to the Polynomial Remainder Theorem, if we have a polynomial P(x) divided by a binomial in the form of (x - a), then the remainder will be given by P(a).

And according to the Factor Theorem, if the remainder of P(x) / (x - a) is zero: that is, if P(a) = 0, then (x - a) is a factor of P(x).

We are given the polynomial:

`P(x)=x^3+6x^2+4x+k`

And we know that it is divisible by:

`(x+2)`

We can rewrite our divisor as (x - (-2)). Hence, a = -2.

According to both the PRT and Factor Theorem, P(-2) must equal 0. Hence:

`P(-2)=0`

Substitute:

`(-2)^3+6(-2)^2+4(-2)+k=0`

Solve for k. Simplify:

`(-8)+6(4)-8+k=0`

Hence:

`k=-8`