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.