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Answer :
Sure! Let's factor the polynomial [tex]\( x^3 - 5x^2 + 9x - 45 \)[/tex] by grouping. Here’s a step-by-step breakdown:
1. Group the terms:
Start by grouping the terms into two pairs.
[tex]\[
(x^3 - 5x^2) + (9x - 45)
\][/tex]
2. Factor each group:
- Look at the first group [tex]\((x^3 - 5x^2)\)[/tex]. You can factor out the greatest common factor, which is [tex]\(x^2\)[/tex].
[tex]\[
x^2(x - 5)
\][/tex]
- Now look at the second group [tex]\((9x - 45)\)[/tex]. The greatest common factor here is 9.
[tex]\[
9(x - 5)
\][/tex]
3. Combine the factored groups:
Now, you can see that both terms have a common factor of [tex]\((x - 5)\)[/tex].
[tex]\[
x^2(x - 5) + 9(x - 5) = (x^2 + 9)(x - 5)
\][/tex]
So the factored form of the polynomial [tex]\( x^3 - 5x^2 + 9x - 45 \)[/tex] is [tex]\((x - 5)(x^2 + 9)\)[/tex].
Therefore, the correct choice is:
A. [tex]\( x^3 - 5x^2 + 9x - 45 = (x - 5)(x^2 + 9) \)[/tex]
1. Group the terms:
Start by grouping the terms into two pairs.
[tex]\[
(x^3 - 5x^2) + (9x - 45)
\][/tex]
2. Factor each group:
- Look at the first group [tex]\((x^3 - 5x^2)\)[/tex]. You can factor out the greatest common factor, which is [tex]\(x^2\)[/tex].
[tex]\[
x^2(x - 5)
\][/tex]
- Now look at the second group [tex]\((9x - 45)\)[/tex]. The greatest common factor here is 9.
[tex]\[
9(x - 5)
\][/tex]
3. Combine the factored groups:
Now, you can see that both terms have a common factor of [tex]\((x - 5)\)[/tex].
[tex]\[
x^2(x - 5) + 9(x - 5) = (x^2 + 9)(x - 5)
\][/tex]
So the factored form of the polynomial [tex]\( x^3 - 5x^2 + 9x - 45 \)[/tex] is [tex]\((x - 5)(x^2 + 9)\)[/tex].
Therefore, the correct choice is:
A. [tex]\( x^3 - 5x^2 + 9x - 45 = (x - 5)(x^2 + 9) \)[/tex]
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