Question

Solve the polynomial equation p(x)=x3+2x2−5x−6 . Which of the following is a factor?(1 point)
Responses

x−3
x minus 3

x+2
x plus 2

x−1
x minus 1

x+1

Answer 1

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 if \( x - c \) is a factor of a polynomial \( p(x) \), then \( p(c) = 0 \).

We will evaluate \( p(x) \) at the values corresponding to each option:

1. **For \( x - 3 \)**, evaluate \( p(3) \):
\[
p(3) = 3^3 + 2(3^2) - 5(3) - 6 = 27 + 18 - 15 - 6 = 24 \quad \text{(not a factor)}
\]

2. **For \( x + 2 \)**, evaluate \( p(-2) \):
\[
p(-2) = (-2)^3 + 2(-2)^2 - 5(-2) - 6 = -8 + 8 + 10 - 6 = 4 \quad \text{(not a factor)}
\]

3. **For \( x - 1 \)**, evaluate \( p(1) \):
\[
p(1) = 1^3 + 2(1^2) - 5(1) - 6 = 1 + 2 - 5 - 6 = -8 \quad \text{(not a factor)}
\]

4. **For \( x + 1 \)**, evaluate \( p(-1) \):
\[
p(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0 \quad \text{(this is a factor)}
\]

Since \( p(-1) = 0 \), the factor corresponding to \( x + 1 \) is indeed a factor of the polynomial \( p(x) \).

Thus, the answer is:

**\( x + 1 \)**