Answer :
Sure, let's break this down step-by-step:
We start with the expression:
[tex]\[ 6x^3 + 48 \][/tex]
1. Identify the Greatest Common Factor (GCF):
We need to factor out the common term from the expression. To do this, we first find the greatest common factor (GCF) of the coefficients and the constant term. In this case, both terms 6 and 48 have a common factor, which is 6.
2. Factor out the GCF:
Once we have identified the GCF, we factor it out from each term in the expression:
[tex]\[
6x^3 + 48 = 6(x^3) + 6(8)
\][/tex]
Now, factor out the 6:
[tex]\[
6(x^3 + 8)
\][/tex]
3. Consider special factoring forms:
Notice that [tex]\(x^3 + 8\)[/tex] is a sum of cubes. The sum of cubes formula is given by:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
Here, [tex]\(x^3\)[/tex] is [tex]\(a^3\)[/tex] and [tex]\(8\)[/tex] is [tex]\(2^3\)[/tex]. Therefore, [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex].
4. Apply the sum of cubes formula:
Substitute [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex] into the sum of cubes formula:
[tex]\[
(x)^3 + (2)^3 = (x + 2)((x)^2 - (x)(2) + (2)^2)
\][/tex]
Simplify inside the parentheses:
[tex]\[
(x + 2)(x^2 - 2x + 4)
\][/tex]
5. Combine everything back together:
Now, bring back the factored out GCF:
[tex]\[
6(x + 2)(x^2 - 2x + 4)
\][/tex]
So, the factorization of the original expression:
[tex]\[ 6x^3 + 48 \][/tex]
is:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
This is the simplified, factored form of the given expression.
We start with the expression:
[tex]\[ 6x^3 + 48 \][/tex]
1. Identify the Greatest Common Factor (GCF):
We need to factor out the common term from the expression. To do this, we first find the greatest common factor (GCF) of the coefficients and the constant term. In this case, both terms 6 and 48 have a common factor, which is 6.
2. Factor out the GCF:
Once we have identified the GCF, we factor it out from each term in the expression:
[tex]\[
6x^3 + 48 = 6(x^3) + 6(8)
\][/tex]
Now, factor out the 6:
[tex]\[
6(x^3 + 8)
\][/tex]
3. Consider special factoring forms:
Notice that [tex]\(x^3 + 8\)[/tex] is a sum of cubes. The sum of cubes formula is given by:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
Here, [tex]\(x^3\)[/tex] is [tex]\(a^3\)[/tex] and [tex]\(8\)[/tex] is [tex]\(2^3\)[/tex]. Therefore, [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex].
4. Apply the sum of cubes formula:
Substitute [tex]\(a = x\)[/tex] and [tex]\(b = 2\)[/tex] into the sum of cubes formula:
[tex]\[
(x)^3 + (2)^3 = (x + 2)((x)^2 - (x)(2) + (2)^2)
\][/tex]
Simplify inside the parentheses:
[tex]\[
(x + 2)(x^2 - 2x + 4)
\][/tex]
5. Combine everything back together:
Now, bring back the factored out GCF:
[tex]\[
6(x + 2)(x^2 - 2x + 4)
\][/tex]
So, the factorization of the original expression:
[tex]\[ 6x^3 + 48 \][/tex]
is:
[tex]\[ 6(x + 2)(x^2 - 2x + 4) \][/tex]
This is the simplified, factored form of the given expression.
