Bot GPT 4-o mini To determine which of the given options is a factor of the polynomial \( p(x) = x^3 + 2x^2 - 5x - 6 \), we can use the Factor Theorem. This theorem states that \( (x - r) \) is a factor of \( p(x) \) if and only if \( p(r) = 0 \).
We will evaluate \( p(x) \) at the possible roots derived from the answer choices:
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For \( x + 1 \) (i.e., \( r = -1 \)): \[ p(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0 \] Since \( p(-1) = 0 \), \( x + 1 \) is a factor of \( p(x) \).
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For \( x - 3 \) (i.e., \( r = 3 \)): \[ p(3) = 3^3 + 2(3)^2 - 5(3) - 6 = 27 + 18 - 15 - 6 = 24 \] \( p(3) \neq 0 \), so \( x - 3 \) is not a factor.
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For \( x + 2 \) (i.e., \( r = -2 \)): \[ p(-2) = (-2)^3 + 2(-2)^2 - 5(-2) - 6 = -8 + 8 + 10 - 6 = 4 \] \( p(-2) \neq 0 \), so \( x + 2 \) is not a factor.
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For \( x - 1 \) (i.e., \( r = 1 \)): \[ p(1) = 1^3 + 2(1)^2 - 5(1) - 6 = 1 + 2 - 5 - 6 = -8 \] \( p(1) \neq 0 \), so \( x - 1 \) is not a factor.
Based on the evaluations, the only factor of the polynomial \( p(x) \) is:
\( x + 1 \).
3 answers