Answer :
To factor the polynomial [tex]\(6x^5 - 48x^3\)[/tex] completely using the greatest common factor (GCF), follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients in the polynomial are 6 and -48.
- The greatest common factor of 6 and 48 is 6.
2. Identify the greatest common factor of the variable terms:
- The variable terms are [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex].
- The greatest common factor of [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex] is [tex]\(x^3\)[/tex], since [tex]\(x^3\)[/tex] is the highest power of [tex]\(x\)[/tex] that divides both [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex].
3. Combine the GCF of the coefficients and the variable terms:
- The overall greatest common factor of the polynomial is [tex]\(6x^3\)[/tex].
4. Factor out the GCF from the polynomial:
- Express each term as a product of the GCF and another factor:
[tex]\[
6x^5 = 6x^3 \cdot x^2
\][/tex]
[tex]\[
-48x^3 = 6x^3 \cdot (-8)
\][/tex]
- Rewrite the polynomial by factoring out [tex]\(6x^3\)[/tex]:
[tex]\[
6x^5 - 48x^3 = 6x^3(x^2) - 6x^3(8)
\][/tex]
5. Simplify the expression inside the parentheses:
- Combine the terms inside the parentheses:
[tex]\[
6x^3(x^2 - 8)
\][/tex]
Therefore, the polynomial [tex]\(6x^5 - 48x^3\)[/tex] factored completely using the greatest common factor is:
[tex]\[
6x^3(x^2 - 8)
\][/tex]
1. Identify the GCF of the coefficients:
- The coefficients in the polynomial are 6 and -48.
- The greatest common factor of 6 and 48 is 6.
2. Identify the greatest common factor of the variable terms:
- The variable terms are [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex].
- The greatest common factor of [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex] is [tex]\(x^3\)[/tex], since [tex]\(x^3\)[/tex] is the highest power of [tex]\(x\)[/tex] that divides both [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex].
3. Combine the GCF of the coefficients and the variable terms:
- The overall greatest common factor of the polynomial is [tex]\(6x^3\)[/tex].
4. Factor out the GCF from the polynomial:
- Express each term as a product of the GCF and another factor:
[tex]\[
6x^5 = 6x^3 \cdot x^2
\][/tex]
[tex]\[
-48x^3 = 6x^3 \cdot (-8)
\][/tex]
- Rewrite the polynomial by factoring out [tex]\(6x^3\)[/tex]:
[tex]\[
6x^5 - 48x^3 = 6x^3(x^2) - 6x^3(8)
\][/tex]
5. Simplify the expression inside the parentheses:
- Combine the terms inside the parentheses:
[tex]\[
6x^3(x^2 - 8)
\][/tex]
Therefore, the polynomial [tex]\(6x^5 - 48x^3\)[/tex] factored completely using the greatest common factor is:
[tex]\[
6x^3(x^2 - 8)
\][/tex]
