Certainly! Let's determine the factors of the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] by grouping.
1. Grouping Terms:
We start by grouping the terms in pairs to factor out common factors:
[tex]\[
x^3 - 9x^2 + 5x - 45
\][/tex]
Grouping the first two terms and the last two terms, we get:
[tex]\[
(x^3 - 9x^2) + (5x - 45)
\][/tex]
2. Factoring Within Each Group:
Now, we factor out the greatest common factor from each group:
- For the first group [tex]\((x^3 - 9x^2)\)[/tex], the common factor is [tex]\(x^2\)[/tex]:
[tex]\[
x^3 - 9x^2 = x^2(x - 9)
\][/tex]
- For the second group [tex]\((5x - 45)\)[/tex], the common factor is 5:
[tex]\[
5x - 45 = 5(x - 9)
\][/tex]
So we rewrite the grouped expression as:
[tex]\[
x^2(x - 9) + 5(x - 9)
\][/tex]
3. Factoring Out the Common Binomial Factor:
We notice that both terms now include a common binomial factor [tex]\((x - 9)\)[/tex]. We can factor out [tex]\((x - 9)\)[/tex] from the entire expression:
[tex]\[
x^2(x - 9) + 5(x - 9) = (x - 9)(x^2 + 5)
\][/tex]
Thus, one way to determine the factors of [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] by grouping is to rewrite it as [tex]\(x^2(x - 9) + 5(x - 9)\)[/tex].
Therefore, the correct answer is:
[tex]\[
x^2(x-9)-5(x-9)
\][/tex]
