Question

The Fundamental Theorem of Algebra Quick Check
4 of 54 of 5 Items
Question
Solve the polynomial equation p(x)=x3+2x2−5x−6
. Which of the following is a factor?(1 point)
Responses

x+2
x plus 2

x−3
x minus 3

x+1
x plus 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, which states that if \( p(c) = 0 \) for some constant \( c \), then \( (x - c) \) is a factor of the polynomial.

Let's evaluate \( p(x) \) at the roots corresponding to the provided options:

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

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

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

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

Therefore, the option that is a factor is:

**\( x + 1 \)**