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Answer :
To factor the expression [tex]\(x^3 - 8x^2 + 6x - 48\)[/tex] by grouping, follow these steps:
1. Group the terms:
We divide the expression into two groups:
- First group: [tex]\(x^3 - 8x^2\)[/tex]
- Second group: [tex]\(6x - 48\)[/tex]
2. Factor out the greatest common factor in each group:
- For the first group [tex]\(x^3 - 8x^2\)[/tex], the greatest common factor is [tex]\(x^2\)[/tex]. Factor it out:
[tex]\[
x^3 - 8x^2 = x^2(x - 8)
\][/tex]
- For the second group [tex]\(6x - 48\)[/tex], the greatest common factor is [tex]\(6\)[/tex]. Factor it out:
[tex]\[
6x - 48 = 6(x - 8)
\][/tex]
3. Notice a common factor:
After factoring, you'll notice that both groups contain a common factor of [tex]\((x - 8)\)[/tex]:
[tex]\[
x^2(x - 8) + 6(x - 8)
\][/tex]
4. Factor out the common binomial:
Factor out [tex]\((x - 8)\)[/tex] from the entire expression:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
Thus, the expression [tex]\(x^3 - 8x^2 + 6x - 48\)[/tex] can be factored by grouping into:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
This is your factored expression.
1. Group the terms:
We divide the expression into two groups:
- First group: [tex]\(x^3 - 8x^2\)[/tex]
- Second group: [tex]\(6x - 48\)[/tex]
2. Factor out the greatest common factor in each group:
- For the first group [tex]\(x^3 - 8x^2\)[/tex], the greatest common factor is [tex]\(x^2\)[/tex]. Factor it out:
[tex]\[
x^3 - 8x^2 = x^2(x - 8)
\][/tex]
- For the second group [tex]\(6x - 48\)[/tex], the greatest common factor is [tex]\(6\)[/tex]. Factor it out:
[tex]\[
6x - 48 = 6(x - 8)
\][/tex]
3. Notice a common factor:
After factoring, you'll notice that both groups contain a common factor of [tex]\((x - 8)\)[/tex]:
[tex]\[
x^2(x - 8) + 6(x - 8)
\][/tex]
4. Factor out the common binomial:
Factor out [tex]\((x - 8)\)[/tex] from the entire expression:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
Thus, the expression [tex]\(x^3 - 8x^2 + 6x - 48\)[/tex] can be factored by grouping into:
[tex]\[
(x - 8)(x^2 + 6)
\][/tex]
This is your factored expression.
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