We must find the value k such that Q(x) = (x+2) is a factor of the cubic polynomial:
[tex]P(x)=x^3-6x^2-11x+k[/tex]
Now, if (x+2) is a factor of the polynomial P(x), then the rest R of the following division must be zero,
[tex]\frac{P(x)}{Q(x)}[/tex]
So we must compute the division between the polynomials and check the condition to have R = 0. We compute the quotient by applying the method of synthetic division. Doing that we have:
From the division we see that the rest is:
[tex]R=k-10[/tex]
The condition is that the rest R must be equal to zero, so:
[tex]R=k-10=0\Rightarrow k=10[/tex]
Answer: k = 10
