Use the Factor Theorem to determine whether $$ x-3 $$ is a factor of $$ P(x)=-x^{4}+2 x^{3}+4 x^{2}-9 $$. Specifically, evaluate $$ P $$ at the proper value, and then determine whether $$ x-3 $$ is a factor. $$ P(\square)=\square $$ $$ \begin{array}{l}x-3 ext { is a factor of } P(x) \ x-3 ext { is not a factor of } P(x)\end{array} $$

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Use the Factor Theorem to determine whether $$ x-3 $$ is a factor of $$ P(x)=-x^{4}+2 x^{3}+4 x^{2}-9 $$. Specifically, evaluate $$ P $$ at the proper value, and then determine whether $$ x-3 $$ is a factor. $$ P(\square)=\square $$ $$ \begin{array}{l}x-3 	ext { is a factor of } P(x) \ x-3 	ext { is not a factor of } P(x)\end{array} $$

UpStudy AI · Accepted Answer

We need to check if $$ x-3 $$ is a factor of $$ P(x)=-x^4+2x^3+4x^2-9 $$ by evaluating the polynomial at $$ x=3 $$.

1. Substitute $$ x=3 $$ into $$ P(x) $$:
   $$
   P(3) = -3^4 + 2(3)^3 + 4(3)^2 - 9
   $$

2. Compute each term:
   $$
   -3^4 = -81,\quad 2(3)^3 = 54,\quad 4(3)^2 = 36
   $$

3. Sum the computed values:
   $$
   P(3) = -81 + 54 + 36 - 9 = 0
   $$

4. Since $$ P(3)=0 $$, by the Factor Theorem, $$ x-3 $$ is a factor of $$ P(x) $$.

Thus,

$$ P(3)=0 $$

$$ x-3 \text{ is a factor of } P(x) $$
$$ P(3) = 0 $$ $$ x-3 $$ is 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.

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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.