rewriting a polynomial in factored form: 2. ( g(x) = x^4 + 2x^3 - 13x^2 + mx + 6 ). if ( (x + 1) ) is a factor of the polynomial ( g(x) ), what is the value of ( m )? explain how you know.

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Accepted Answer

# Explanation:
## Step1: Recall the Factor Theorem
The Factor Theorem states that if $(x - a)$ is a factor of a polynomial $G(x)$, then $G(a)=0$. Here, the factor is $(x + 1)$, which can be written as $(x - (-1))$, so $a=-1$.

## Step2: Substitute $x = -1$ into $G(x)$
We have $G(x)=x^{4}+2x^{3}-13x^{2}+mx + 6$. Substitute $x=-1$ into $G(x)$:
$$
\begin{align*}
G(-1)&=(-1)^{4}+2(-1)^{3}-13(-1)^{2}+m(-1)+6\
&=1+2(-1)-13(1)-m + 6
\end{align*}
$$

## Step3: Simplify the expression
Simplify each term:
$$
\begin{align*}
G(-1)&=1-2 - 13 - m+6\
&=(1 + 6)-(2 + 13)-m\
&=7 - 15 - m\
&=-8 - m
\end{align*}
$$

## Step4: Set $G(-1)=0$ and solve for $m$
Since $(x + 1)$ is a factor, $G(-1)=0$. So:
$$
\begin{align*}
-8 - m&=0\
-m&=8\
m&=-8
\end{align*}
$$

# Answer:
The value of $m$ is $-8$.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the Factor Theorem

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

Step3: Simplify the expression

Step4: Set \(G(-1)=0\) and solve for \(m\)

Answer: