College

Factor the GCF out of the polynomial below:

[tex] 20x^6 + 12x^4 + 8x^3 [/tex]

Include both the GCF and the remaining polynomial in your solution.

Answer :

To factor out the greatest common factor (GCF) from the polynomial [tex]\(20x^6 + 12x^4 + 8x^3\)[/tex], follow these steps:

1. Identify the GCF of the coefficients:
- The coefficients of the polynomial are 20, 12, and 8.
- The greatest common factor (GCF) of these numbers is 4.

2. Identify the GCF of the variable parts:
- Each term in the polynomial has a variable part with [tex]\(x\)[/tex]: [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex].
- The smallest power of [tex]\(x\)[/tex] present in all terms is [tex]\(x^3\)[/tex].

3. Combine the GCF of the coefficients and the variable parts:
- The GCF for the whole polynomial expression is [tex]\(4x^3\)[/tex].

4. Factor out the GCF:
- Divide each term by the GCF [tex]\(4x^3\)[/tex]:
- [tex]\(20x^6 \div 4x^3 = 5x^3\)[/tex]
- [tex]\(12x^4 \div 4x^3 = 3x\)[/tex]
- [tex]\(8x^3 \div 4x^3 = 2\)[/tex]

5. Write the factored expression:
- After factoring out [tex]\(4x^3\)[/tex], the polynomial can be written as:
[tex]\[
20x^6 + 12x^4 + 8x^3 = 4x^3(5x^3 + 3x + 2)
\][/tex]

So, the GCF is [tex]\(4x^3\)[/tex] and the remaining polynomial after factoring out the GCF is [tex]\(5x^3 + 3x + 2\)[/tex].

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