difficulty: ★★★★☆ my answer one of the factors of $x^3 + 2x^2 + 2x + k$ is $x + 1$ where $k$ is a constant, find the value of $k$.

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Accepted Answer

# Explanation:
## Step1: Recall Factor Theorem
If $x + a$ is a factor of a polynomial $f(x)$, then $f(-a)=0$. Here, the factor is $x + 1$, so $a = 1$ and we need to find $f(-1)=0$ where $f(x)=x^{3}+2x^{2}+2x + k$.

## Step2: Substitute $x=-1$ into $f(x)$
Substitute $x=-1$ into $f(x)=x^{3}+2x^{2}+2x + k$:
$$
\begin{align*}
f(-1)&=(-1)^{3}+2\times(-1)^{2}+2\times(-1)+k\
&=-1 + 2\times1-2 + k\
&=-1 + 2-2 + k
\end{align*}
$$

## Step3: Solve for $k$
Since $f(-1) = 0$ (by Factor Theorem):
$$
\begin{align*}
-1 + 2-2 + k&=0\
(-1 + 2-2)+k&=0\
(-1)+k&=0\
k&=1
\end{align*}
$$

# Answer:
$k = 1$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Factor Theorem

Step2: Substitute \(x=-1\) into \(f(x)\)

Step3: Solve for \(k\)

Answer: