Question

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.

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 \(-8\).