Answer:
[tex]k=-8[/tex]
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:
[tex]P(x)=x^3+6x^2+4x+k[/tex]
And we know that it is divisible by:
[tex](x+2)[/tex]
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:
[tex]P(-2)=0[/tex]
Substitute:
[tex](-2)^3+6(-2)^2+4(-2)+k=0[/tex]
Solve for k. Simplify:
[tex](-8)+6(4)-8+k=0[/tex]
Hence:
[tex]k=-8[/tex]
