Answer :

Sure! Let's go through the process of factoring out the greatest common factor step-by-step:

1. Identify the Terms:
- The expression given is [tex]\(3x^7 + 12x^5 + 9x^2\)[/tex].

2. Find the Greatest Common Factor (GCF):
- First, look at the coefficients: 3, 12, and 9. The greatest common factor of these numbers is 3.
- Now, look at the variable part. Each term has [tex]\(x\)[/tex] raised to a power. The smallest exponent is 2 (from [tex]\(9x^2\)[/tex]), so the greatest common factor for the variable part is [tex]\(x^2\)[/tex].

3. Factor Out the GCF:
- Since the greatest common factor of the entire expression is [tex]\(3x^2\)[/tex], we factor this out from each term:
- [tex]\(3x^7\)[/tex] becomes [tex]\(3x^2 \cdot x^5\)[/tex],
- [tex]\(12x^5\)[/tex] becomes [tex]\(3x^2 \cdot 4x^3\)[/tex],
- [tex]\(9x^2\)[/tex] becomes [tex]\(3x^2 \cdot 3\)[/tex].

4. Rewrite the Expression:
- Factoring [tex]\(3x^2\)[/tex] out of each term, the expression becomes:
[tex]\[
3x^2(x^5 + 4x^3 + 3)
\][/tex]

So, the expression [tex]\(3x^7 + 12x^5 + 9x^2\)[/tex] factors to [tex]\(3x^2(x^5 + 4x^3 + 3)\)[/tex].

This step-by-step process shows how to factor out the greatest common factor from the given polynomial.

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