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Answer :
To factor the polynomial [tex]\(x^3 - 6x^2 + 8x - 48\)[/tex] by grouping, we can follow these steps:
1. Group the terms:
We'll split the polynomial into two groups:
[tex]\[
(x^3 - 6x^2) + (8x - 48)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\((x^3 - 6x^2)\)[/tex]:
The GCF is [tex]\(x^2\)[/tex], so we factor it out:
[tex]\[
x^2(x - 6)
\][/tex]
- From the second group [tex]\((8x - 48)\)[/tex]:
The GCF is 8, so we factor it out:
[tex]\[
8(x - 6)
\][/tex]
3. Write the expression with these factors:
Now we have:
[tex]\[
x^2(x - 6) + 8(x - 6)
\][/tex]
4. Factor the common binomial:
Notice that both terms have a common factor of [tex]\((x - 6)\)[/tex]. We can factor that out:
[tex]\[
(x^2 + 8)(x - 6)
\][/tex]
Thus, the polynomial [tex]\(x^3 - 6x^2 + 8x - 48\)[/tex] can be factored as:
[tex]\[
(x - 6)(x^2 + 8)
\][/tex]
Therefore, the correct choice is:
A. [tex]\(x^3 - 6x^2 + 8x - 48 = (x - 6)(x^2 + 8)\)[/tex]
1. Group the terms:
We'll split the polynomial into two groups:
[tex]\[
(x^3 - 6x^2) + (8x - 48)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\((x^3 - 6x^2)\)[/tex]:
The GCF is [tex]\(x^2\)[/tex], so we factor it out:
[tex]\[
x^2(x - 6)
\][/tex]
- From the second group [tex]\((8x - 48)\)[/tex]:
The GCF is 8, so we factor it out:
[tex]\[
8(x - 6)
\][/tex]
3. Write the expression with these factors:
Now we have:
[tex]\[
x^2(x - 6) + 8(x - 6)
\][/tex]
4. Factor the common binomial:
Notice that both terms have a common factor of [tex]\((x - 6)\)[/tex]. We can factor that out:
[tex]\[
(x^2 + 8)(x - 6)
\][/tex]
Thus, the polynomial [tex]\(x^3 - 6x^2 + 8x - 48\)[/tex] can be factored as:
[tex]\[
(x - 6)(x^2 + 8)
\][/tex]
Therefore, the correct choice is:
A. [tex]\(x^3 - 6x^2 + 8x - 48 = (x - 6)(x^2 + 8)\)[/tex]
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