To determine the factors of the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] by grouping, let's follow the steps for factoring by grouping.
First, we need to group the terms in pairs in such a way that we can factor out a common factor from each group. Here is the polynomial:
[tex]\[ x^3 - 9x^2 + 5x - 45 \][/tex]
1. Group the terms:
Group the polynomial into two pairs:
[tex]\[ (x^3 - 9x^2) + (5x - 45) \][/tex]
2. Factor out the greatest common factor from each pair:
- For the first group [tex]\(x^3 - 9x^2\)[/tex], factor out [tex]\(x^2\)[/tex]:
[tex]\[ x^2(x - 9) \][/tex]
- For the second group [tex]\(5x - 45\)[/tex], factor out [tex]\(5\)[/tex]:
[tex]\[ 5(x - 9) \][/tex]
3. Rewrite the expression showing the grouped pairs:
[tex]\[ x^2(x - 9) + 5(x - 9) \][/tex]
4. Factor out the common binomial factor [tex]\((x - 9)\)[/tex]:
Notice that [tex]\((x - 9)\)[/tex] is a common factor in both terms:
[tex]\[ (x^2 + 5)(x - 9) \][/tex]
Therefore, the factors of the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] by grouping are [tex]\((x^2 + 5)(x - 9)\)[/tex].
So, the correct choice from the given options is:
[tex]\[ x^2(x - 9) - 5(x - 9) \][/tex]
