Questions: Solve the problem. You must show your work. Clearly label your answer. Use the factor theorem to decide whether x+3 is a factor of 7x^4+22x^3-3x^2+x+4.

Solve the problem. You must show your work. Clearly label your answer. Use the factor theorem to decide whether x+3 is a factor of 7x^4+22x^3-3x^2+x+4.

Transcript text: Solve the problem. You must show your work. Clearly label your answer. Use the factor theorem to decide whether $\mathrm{x}+3$ is a factor of $7 x^{4}+22 x^{3}-3 x^{2}+x+4$.

Solution

Solution Steps

To determine if $x+3$ is a factor of the polynomial $7x^4 + 22x^3 - 3x^2 + x + 4$ , we can use the factor theorem. According to the factor theorem, $x+3$ is a factor of the polynomial if and only if substituting $x = -3$ into the polynomial results in zero. Therefore, we will evaluate the polynomial at $x = -3$ .

Step 1: Evaluate the Polynomial

To determine if $x + 3$ is a factor of the polynomial $7x^4 + 22x^3 - 3x^2 + x + 4$ , we substitute $x = -3$ into the polynomial:

$P(-3) = 7(-3)^4 + 22(-3)^3 - 3(-3)^2 + (-3) + 4$

Calculating each term:

$7(-3)^4 = 7 \cdot 81 = 567$
$22(-3)^3 = 22 \cdot (-27) = -594$
$-3(-3)^2 = -3 \cdot 9 = -27$
$-3 = -3$
$4 = 4$

Now, summing these results:

$P(-3) = 567 - 594 - 27 - 3 + 4 = -53$

Step 2: Apply the Factor Theorem

According to the factor theorem, $x + 3$ is a factor of the polynomial if $P(-3) = 0$ . Since we found that $P(-3) = -53$ , which is not equal to zero, we conclude that $x + 3$ is not a factor of the polynomial.