To determine which of the provided options is a factor of the polynomial \( p(x) = x^3 + 2x^2 - 5x - 6 \), we can use the Factor Theorem. According to the theorem, if \( x - c \) is a factor of \( p(x) \), then \( p(c) = 0 \).
Let’s evaluate \( p(x) \) at the potential roots given by the factors:
-
For \( x - 1 \) (i.e., \( c = 1 \)): \[ p(1) = 1^3 + 2(1^2) - 5(1) - 6 = 1 + 2 - 5 - 6 = -8 \quad \text{(Not a root)} \]
-
For \( x + 2 \) (i.e., \( c = -2 \)): \[ p(-2) = (-2)^3 + 2(-2)^2 - 5(-2) - 6 = -8 + 8 + 10 - 6 = 4 \quad \text{(Not a root)} \]
-
For \( x - 3 \) (i.e., \( c = 3 \)): \[ p(3) = 3^3 + 2(3^2) - 5(3) - 6 = 27 + 18 - 15 - 6 = 24 \quad \text{(Not a root)} \]
-
For \( x + 1 \) (i.e., \( c = -1 \)): \[ p(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0 \quad \text{(This is a root)} \]
Since \( p(-1) = 0 \), \( x + 1 \) is a factor of the polynomial \( p(x) \).
Thus, the correct response is:
- \( x + 1 \)