Question

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 $k$.

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:

\(k = 1\)