Problem

11 Find the values of $h$ and $k$ given that $x+2$ is a factor of $Q(x)=(x+h)^{2}+k$, and the remainder is 16 when $Q(x)$ is divided by $x$.

Answer

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Answer

\(\boxed{h = 5}\) and \(\boxed{k = -9}\)

Steps

Step 1 ：Given that $x+2$ is a factor of $Q(x)=(x+h)^{2}+k$, we have $Q(-2) = 0$.

Step 2 ：Substitute $x=-2$ into the equation: $Q(-2) = (-2+h)^2 + k = 0$.

Step 3 ：Given that the remainder is 16 when $Q(x)$ is divided by $x$, we have $Q(0) = 16$.

Step 4 ：Substitute $x=0$ into the equation: $Q(0) = h^2 + k = 16$.

Step 5 ：Now we have a system of equations: \begin{cases} (-2+h)^2 + k = 0 \\ h^2 + k = 16 \end{cases}

Step 6 ：Solve the system of equations to find the values of $h$ and $k$.

Step 7 ：\(\boxed{h = 5}\) and \(\boxed{k = -9}\)

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