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Answer :
To factor the polynomial [tex]\(x^3 + 6x^2 + 8x + 48\)[/tex] completely, we can follow these steps:
1. Group Terms:
Start by grouping the terms to see if we can factor them by grouping:
[tex]\[
(x^3 + 6x^2) + (8x + 48)
\][/tex]
2. Factor Each Group:
- For the first group [tex]\(x^3 + 6x^2\)[/tex], factor out the greatest common factor (GCF), which is [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x + 6)
\][/tex]
- For the second group [tex]\(8x + 48\)[/tex], factor out the GCF, which is 8:
[tex]\[
8(x + 6)
\][/tex]
3. Combine and Factor Further:
Now the expression looks like this:
[tex]\[
x^2(x + 6) + 8(x + 6)
\][/tex]
Notice that both terms have a common factor of [tex]\((x + 6)\)[/tex]. We can factor [tex]\((x + 6)\)[/tex] out:
[tex]\[
(x + 6)(x^2 + 8)
\][/tex]
4. Examine the Remaining Factor:
The remaining factor is [tex]\(x^2 + 8\)[/tex]. This expression is already simplified and cannot be factored further with real numbers.
Therefore, the polynomial [tex]\(x^3 + 6x^2 + 8x + 48\)[/tex] factors completely to:
[tex]\[
(x + 6)(x^2 + 8)
\][/tex]
1. Group Terms:
Start by grouping the terms to see if we can factor them by grouping:
[tex]\[
(x^3 + 6x^2) + (8x + 48)
\][/tex]
2. Factor Each Group:
- For the first group [tex]\(x^3 + 6x^2\)[/tex], factor out the greatest common factor (GCF), which is [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x + 6)
\][/tex]
- For the second group [tex]\(8x + 48\)[/tex], factor out the GCF, which is 8:
[tex]\[
8(x + 6)
\][/tex]
3. Combine and Factor Further:
Now the expression looks like this:
[tex]\[
x^2(x + 6) + 8(x + 6)
\][/tex]
Notice that both terms have a common factor of [tex]\((x + 6)\)[/tex]. We can factor [tex]\((x + 6)\)[/tex] out:
[tex]\[
(x + 6)(x^2 + 8)
\][/tex]
4. Examine the Remaining Factor:
The remaining factor is [tex]\(x^2 + 8\)[/tex]. This expression is already simplified and cannot be factored further with real numbers.
Therefore, the polynomial [tex]\(x^3 + 6x^2 + 8x + 48\)[/tex] factors completely to:
[tex]\[
(x + 6)(x^2 + 8)
\][/tex]
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