To factor the polynomial [tex]\( x^3 - 5x^2 + x - 5 \)[/tex], follow these steps:
1. Group the terms in pairs:
[tex]\[
(x^3 - 5x^2) + (x - 5)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- In the first group, [tex]\( x^3 - 5x^2 \)[/tex], the GCF is [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x - 5)
\][/tex]
- In the second group, [tex]\( x - 5 \)[/tex], there is no common factor other than 1:
[tex]\[
1(x - 5)
\][/tex]
3. Rewrite the polynomial with the factored terms:
[tex]\[
x^2(x - 5) + 1(x - 5)
\][/tex]
4. Factor out the common binomial factor [tex]\((x - 5)\)[/tex]:
[tex]\[
(x - 5)(x^2 + 1)
\][/tex]
Therefore, the polynomial [tex]\( x^3 - 5x^2 + x - 5 \)[/tex] factors as follows:
[tex]\[
\boxed{(x - 5)(x^2 + 1)}
\][/tex]
So the correct choice is:
A. [tex]\( x^3 - 5x^2 + x - 5 = (x - 5)(x^2 + 1) \)[/tex]
