Answer :
Sure, let's go through the process of factoring the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] by grouping. Here are the steps:
1. Group the terms in pairs:
[tex]\[
(x^3 - 9x^2) + (5x - 45)
\][/tex]
2. Factor out the greatest common factor (GCF) from each pair:
- From [tex]\(x^3 - 9x^2\)[/tex], the GCF is [tex]\(x^2\)[/tex]:
[tex]\[
x^2 (x - 9)
\][/tex]
- From [tex]\(5x - 45\)[/tex], the GCF is 5:
[tex]\[
5 (x - 9)
\][/tex]
This gives us:
[tex]\[
x^2 (x - 9) + 5 (x - 9)
\][/tex]
3. Factor out the common binomial factor [tex]\((x - 9)\)[/tex]:
[tex]\[
(x - 9) (x^2 + 5)
\][/tex]
So, the factored form of the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] is:
[tex]\[
(x - 9)(x^2 + 5)
\][/tex]
Therefore, the correct choice is:
A. [tex]\(x^3 - 9x^2 + 5x - 45 = (x - 9)(x^2 + 5)\)[/tex]
1. Group the terms in pairs:
[tex]\[
(x^3 - 9x^2) + (5x - 45)
\][/tex]
2. Factor out the greatest common factor (GCF) from each pair:
- From [tex]\(x^3 - 9x^2\)[/tex], the GCF is [tex]\(x^2\)[/tex]:
[tex]\[
x^2 (x - 9)
\][/tex]
- From [tex]\(5x - 45\)[/tex], the GCF is 5:
[tex]\[
5 (x - 9)
\][/tex]
This gives us:
[tex]\[
x^2 (x - 9) + 5 (x - 9)
\][/tex]
3. Factor out the common binomial factor [tex]\((x - 9)\)[/tex]:
[tex]\[
(x - 9) (x^2 + 5)
\][/tex]
So, the factored form of the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] is:
[tex]\[
(x - 9)(x^2 + 5)
\][/tex]
Therefore, the correct choice is:
A. [tex]\(x^3 - 9x^2 + 5x - 45 = (x - 9)(x^2 + 5)\)[/tex]