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

Step-by-step 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:

And we know that it is divisible by:

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:

Substitute:

Solve for k. Simplify:

Hence:

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step-by-step explanation: