Answer :

Sure, let's factor out the Greatest Common Factor (GCF) from the polynomial [tex]\(10x^5 + 35x^3 + 5x^2\)[/tex].

### Step-by-Step Solution:

1. Identify the GCF of the coefficients:
- The coefficients are 10, 35, and 5.
- The GCF of 10, 35, and 5 is 5, since 5 is the largest number that divides all three of them equally.

2. Identify the GCF of the variable terms:
- The variable terms are [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] among these terms is [tex]\(x^2\)[/tex]. Therefore, [tex]\(x^2\)[/tex] is the GCF of the variable terms.

3. Combine the GCF of the coefficients and the variable terms:
- The overall GCF is [tex]\(5x^2\)[/tex].

4. Factor out the GCF from each term in the polynomial:
- Divide each term by [tex]\(5x^2\)[/tex]:
[tex]\[
\frac{10x^5}{5x^2} = 2x^3
\][/tex]
[tex]\[
\frac{35x^3}{5x^2} = 7x
\][/tex]
[tex]\[
\frac{5x^2}{5x^2} = 1
\][/tex]

5. Write the factored form:
- After factoring out [tex]\(5x^2\)[/tex], we get:
[tex]\[
10x^5 + 35x^3 + 5x^2 = 5x^2 (2x^3 + 7x + 1)
\][/tex]

So, the factored form of the polynomial [tex]\(10x^5 + 35x^3 + 5x^2\)[/tex] is:
[tex]\[
5x^2 (2x^3 + 7x + 1)
\][/tex]

This is the fully factored expression.

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