To determine the relationship between [tex]\( (x+1) \)[/tex] and the polynomial [tex]\( x^2 - x - 2 \)[/tex], let's go through the process of factorization step by step.
### Step-by-Step Solution:
1. Write Down the Polynomial:
The given polynomial is:
[tex]\[
P(x) = x^2 - x - 2
\][/tex]
2. Factorize the Polynomial:
We need to factorize the polynomial to see if [tex]\( (x+1) \)[/tex] is one of its factors.
To factorize [tex]\( P(x) \)[/tex], we look for two numbers that multiply to give the constant term ([tex]\(-2\)[/tex]) and add to give the coefficient of the linear term ([tex]\(-1\)[/tex]):
[tex]\[
x^2 - x - 2 = (x + a)(x + b)
\][/tex]
We need:
[tex]\[
a \cdot b = -2 \quad \text{and} \quad a + b = -1
\][/tex]
Trying different pairs, we find that:
[tex]\[
a = -2, \quad b = 1 \quad \text{(or vice versa)}
\][/tex]
So, we can write:
[tex]\[
x^2 - x - 2 = (x - 2)(x + 1)
\][/tex]
3. Identify Factors:
From the factorization, we have:
[tex]\[
x^2 - x - 2 = (x - 2)(x + 1)
\][/tex]
Clearly, [tex]\( (x + 1) \)[/tex] is one of the factors of the polynomial [tex]\( x^2 - x - 2 \)[/tex].
4. Conclusion:
Based on the factorization, we can conclude that [tex]\( (x+1) \)[/tex] is indeed a factor of the polynomial [tex]\( x^2 - x - 2 \)[/tex].
Therefore, the best description of the relationship between [tex]\( (x+1) \)[/tex] and the polynomial [tex]\( x^2 - x - 2 \)[/tex] is:
A. [tex]\( (x+1) \)[/tex] is a factor.
