Answer :
Sure! Let's work through factoring the expression step by step:
We need to factor the expression [tex]\(6x^3 - 36x^2 - 8x + 48\)[/tex].
1. Look for the Greatest Common Factor (GCF):
First, we look for the greatest common factor among all the terms. Here, each coefficient (6, -36, -8, and 48) is divisible by 2. Factoring out the GCF of 2, we get:
[tex]\[
6x^3 - 36x^2 - 8x + 48 = 2(3x^3 - 18x^2 - 4x + 24)
\][/tex]
2. Factor by Grouping:
Now, we focus on factoring what is inside the parentheses. We can try factoring by grouping:
- Group the terms: [tex]\((3x^3 - 18x^2)\)[/tex] and [tex]\((-4x + 24)\)[/tex].
- Factor out the greatest common factor from each group:
- From [tex]\(3x^3 - 18x^2\)[/tex], factor out [tex]\(3x^2\)[/tex]:
[tex]\[
3x^2(x - 6)
\][/tex]
- From [tex]\(-4x + 24\)[/tex], factor out [tex]\(-4\)[/tex]:
[tex]\[
-4(x - 6)
\][/tex]
- We then write:
[tex]\[
3x^2(x - 6) - 4(x - 6)
\][/tex]
3. Factor out the common binomial:
Both terms now contain the common factor [tex]\((x - 6)\)[/tex]. We factor [tex]\((x - 6)\)[/tex] out:
[tex]\[
(3x^2 - 4)(x - 6)
\][/tex]
4. Combine everything:
Don't forget the factor of 2 we took out at the beginning:
[tex]\[
2(3x^2 - 4)(x - 6)
\][/tex]
Thus, the factored form of the expression [tex]\(6x^3 - 36x^2 - 8x + 48\)[/tex] is [tex]\(\boxed{2(x - 6)(3x^2 - 4)}\)[/tex].
We need to factor the expression [tex]\(6x^3 - 36x^2 - 8x + 48\)[/tex].
1. Look for the Greatest Common Factor (GCF):
First, we look for the greatest common factor among all the terms. Here, each coefficient (6, -36, -8, and 48) is divisible by 2. Factoring out the GCF of 2, we get:
[tex]\[
6x^3 - 36x^2 - 8x + 48 = 2(3x^3 - 18x^2 - 4x + 24)
\][/tex]
2. Factor by Grouping:
Now, we focus on factoring what is inside the parentheses. We can try factoring by grouping:
- Group the terms: [tex]\((3x^3 - 18x^2)\)[/tex] and [tex]\((-4x + 24)\)[/tex].
- Factor out the greatest common factor from each group:
- From [tex]\(3x^3 - 18x^2\)[/tex], factor out [tex]\(3x^2\)[/tex]:
[tex]\[
3x^2(x - 6)
\][/tex]
- From [tex]\(-4x + 24\)[/tex], factor out [tex]\(-4\)[/tex]:
[tex]\[
-4(x - 6)
\][/tex]
- We then write:
[tex]\[
3x^2(x - 6) - 4(x - 6)
\][/tex]
3. Factor out the common binomial:
Both terms now contain the common factor [tex]\((x - 6)\)[/tex]. We factor [tex]\((x - 6)\)[/tex] out:
[tex]\[
(3x^2 - 4)(x - 6)
\][/tex]
4. Combine everything:
Don't forget the factor of 2 we took out at the beginning:
[tex]\[
2(3x^2 - 4)(x - 6)
\][/tex]
Thus, the factored form of the expression [tex]\(6x^3 - 36x^2 - 8x + 48\)[/tex] is [tex]\(\boxed{2(x - 6)(3x^2 - 4)}\)[/tex].
