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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:
Begin by grouping the polynomial into two pairs:
[tex]\[
(x^3 + 6x^2) + (-8x - 48)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- In the first group, [tex]\(x^3 + 6x^2\)[/tex], the GCF is [tex]\(x^2\)[/tex]. So, factor it out:
[tex]\[
x^2(x + 6)
\][/tex]
- In the second group, [tex]\(-8x - 48\)[/tex], the GCF is [tex]\(-8\)[/tex]. So, factor it out:
[tex]\[
-8(x + 6)
\][/tex]
3. Combine the factored groups:
Now, notice that both groups contain a common binomial factor, [tex]\((x + 6)\)[/tex]:
[tex]\[
x^2(x + 6) - 8(x + 6)
\][/tex]
4. Factor out the common binomial:
Factor out [tex]\((x + 6)\)[/tex] from the expression:
[tex]\[
(x + 6)(x^2 - 8)
\][/tex]
Now, your original polynomial [tex]\(x^3 + 6x^2 - 8x - 48\)[/tex] is factored as:
[tex]\[
(x + 6)(x^2 - 8)
\][/tex]
This is the factored form of the polynomial using the method of grouping.
1. Group the terms:
Begin by grouping the polynomial into two pairs:
[tex]\[
(x^3 + 6x^2) + (-8x - 48)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- In the first group, [tex]\(x^3 + 6x^2\)[/tex], the GCF is [tex]\(x^2\)[/tex]. So, factor it out:
[tex]\[
x^2(x + 6)
\][/tex]
- In the second group, [tex]\(-8x - 48\)[/tex], the GCF is [tex]\(-8\)[/tex]. So, factor it out:
[tex]\[
-8(x + 6)
\][/tex]
3. Combine the factored groups:
Now, notice that both groups contain a common binomial factor, [tex]\((x + 6)\)[/tex]:
[tex]\[
x^2(x + 6) - 8(x + 6)
\][/tex]
4. Factor out the common binomial:
Factor out [tex]\((x + 6)\)[/tex] from the expression:
[tex]\[
(x + 6)(x^2 - 8)
\][/tex]
Now, your original polynomial [tex]\(x^3 + 6x^2 - 8x - 48\)[/tex] is factored as:
[tex]\[
(x + 6)(x^2 - 8)
\][/tex]
This is the factored form of the polynomial using the method of grouping.
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