To determine which of the given factors, if any, is a factor of the polynomial [tex]\( x^3 + 12x^2 + 9x - 22 \)[/tex], we will use the factor theorem. The factor theorem states that [tex]\( x - a \)[/tex] is a factor of a polynomial [tex]\( P(x) \)[/tex] if and only if [tex]\( P(a) = 0 \)[/tex]. This means we should evaluate the polynomial at the roots associated with the given candidates (i.e., the opposites of [tex]\( a \)[/tex] for each factor [tex]\( x \pm a \)[/tex]) and see if the result is zero.
Let's evaluate the polynomial at the specific values corresponding to each factor:
1. [tex]\( x + 2 \)[/tex]:
- Candidate root: [tex]\( -2 \)[/tex]
- Substitute [tex]\( x = -2 \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[
P(-2) = (-2)^3 + 12(-2)^2 + 9(-2) - 22
= -8 + 48 - 18 - 22
= -8 + 48 - 40
= 0
\][/tex]
Hence, [tex]\( x + 2 \)[/tex] is a factor.
2. [tex]\( x - 2 \)[/tex]:
- Candidate root: [tex]\( 2 \)[/tex]
- Substitute [tex]\( x = 2 \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[
P(2) = 2^3 + 12(2)^2 + 9(2) - 22
= 8 + 48 + 18 - 22
= 8 + 48 + 18 - 22
= 52
\][/tex]
Hence, [tex]\( x - 2 \)[/tex] is not a factor.
3. [tex]\( x - 3 \)[/tex]:
- Candidate root: [tex]\( 3 \)[/tex]
- Substitute [tex]\( x = 3 \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[
P(3) = 3^3 + 12(3)^2 + 9(3) - 22
= 27 + 108 + 27 - 22
= 162 - 22
= 140
\][/tex]
Hence, [tex]\( x - 3 \)[/tex] is not a factor.
4. [tex]\( x + 1 \)[/tex]:
- Candidate root: [tex]\( -1 \)[/tex]
- Substitute [tex]\( x = -1 \)[/tex] into [tex]\( P(x) \)[/tex]
[tex]\[
P(-1) = (-1)^3 + 12(-1)^2 + 9(-1) - 22
= -1 + 12 - 9 - 22
= 2 - 22
= -20
\][/tex]
Hence, [tex]\( x + 1 \)[/tex] is not a factor.
So, the correct answer is:
A. [tex]\( x+2 \)[/tex]
