College

Factor by grouping.

[tex] x^3 - 9x^2 + 5x - 45 [/tex]

Select the correct choice below and fill in any answer boxes with your choice.

A. [tex] x^3 - 9x^2 + 5x - 45 = \square [/tex]

B. The polynomial is prime.

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]

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