\begin{tabular}{l} Use the Factor Theorem to determine whether $$ x+2 $$ is a factor of $$ P(x)=2 x^{4}+x^{3}-4 x-7 $$ \ Specifically, evaluate $$ P $$ at the proper value, and then determine whether $$ x+2 $$ is a factor. \ \hline$$ P(\square)=\square $$ \ \hline$$ x+2 $$ is a factor of $$ P(x) $$ \ \begin{tabular}{l} is not a factor of $$ P(x) $$ \ \hline\end{tabular}\end{tabular}

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UpStudy AI · Accepted Answer

To determine whether $$ x+2 $$ is a factor of $$ P(x)=2x^4+x^3-4x-7 $$ using the Factor Theorem, we need to evaluate $$ P(-2) $$ and check if the result is equal to zero.

The Factor Theorem states that if $$ P(a) = 0 $$, then $$ x - a $$ is a factor of $$ P(x) $$.

Let's substitute $$ x = -2 $$ into the polynomial $$ P(x) $$ and calculate the result.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$2x^{4}+x^{3}-4x-7$$
- step1: Substitute:
  $$2\left(-2\right)^{4}+\left(-2\right)^{3}-4\left(-2\right)-7$$
- step2: Multiply the terms:
  $$2^{5}+\left(-2\right)^{3}-4\left(-2\right)-7$$
- step3: Multiply the numbers:
  $$2^{5}+\left(-2\right)^{3}+8-7$$
- step4: Evaluate the power:
  $$32+\left(-2\right)^{3}+8-7$$
- step5: Evaluate the power:
  $$32-8+8-7$$
- step6: Calculate:
  $$25$$
Substituting $$ x = -2 $$ into the polynomial $$ P(x) $$ gives us $$ P(-2) = 25 $$.

Since $$ P(-2) $$ is not equal to zero, according to the Factor Theorem, $$ x+2 $$ is not a factor of $$ P(x) $$.

Therefore, the correct answer is:

$$ x+2 $$ is not a factor of $$ P(x) $$
$$ x+2 $$ is not a factor of $$ P(x) $$.

Use the Factor Theorem to determine whether is a factor of

Specifically, evaluate at the proper value, and then determine whether is a factor.

is a factor of

is not a factor of

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Use the Factor Theorem to determine whether is a factor of Specifically, evaluate at the proper value, and then determine whether is a factor. is a factor of is not a factor of