Question

use the factor theorem to determine whether x + 2 is a factor of p(x) = x⁴ + 2x³ - 3x - 5. specifically, evaluate p at the proper value, and then determine whether x + 2 is a factor. p(□) = □. options: x + 2 is a factor of p(x); x + 2 is not a factor of p(x)

Explanation:

Step1: Find the value to substitute

The Factor Theorem states that if \( x + a \) is a factor of \( P(x) \), then \( P(-a) = 0 \). For \( x + 2 \), we set \( x + 2 = 0 \), so \( x = -2 \). We need to evaluate \( P(-2) \).

Step2: Substitute \( x = -2 \) into \( P(x) \)

$$
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Since \( P(-2) = 1
eq 0 \), by the Factor Theorem, \( x + 2 \) is not a factor of \( P(x) \).

Answer:

\( P(-2) = 1 \)
\( x + 2 \) is not a factor of \( P(x) \)