<s>[INST] Find k so that polynomial division has remainder 0 
Find $k$ so that $x^3-kx^2+3x+7k$ has remainder $0$ when divided by $x+2$.

How do I approach this problem? I know how to do polynomial long division and synthetic division, but I don't know how to apply it in this equation.
 [/INST] You know the following:
$$
x^3 - kx^2 + 3x + 7k = (x + 2)(ax^2 + bx + c)
$$
Just do the multiplication and find $a$, $b$, and $c$.
$$
(x + 2)(ax^2 + bx + c) = ax^3 + (2a + b)x^2 + (c + 2b)x + 2c
$$
Now we set each coefficient equal:
$$
a = 1 \\
2a + b = -k \rightarrow b = -k - 2\\
c + 2b = 3 \rightarrow c = 3 - 2(-k - 2) \\
2c = 7k \rightarrow 2(3 - 2(-k - 2) = 7k
$$
Solving the last equation gives:
$$
6 + 4k + 8 = 7k \rightarrow 14 = 3k \rightarrow k = \frac{14}{3}
$$
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