Factor out the greatest common factor in the expression:

[tex]7x^4 - 35x^3 + 21x^2[/tex]

Select the correct choice below and fill in any answer boxes within your choice.

A. [tex]7x^4 - 35x^3 + 21x^2 = \square[/tex]

B. The polynomial is prime.

Answer :

Certainly! Let's factor out the greatest common factor (GCF) from the expression [tex]\(7x^4 - 35x^3 + 21x^2\)[/tex].

### Step-by-Step Solution:

1. Identify the GCF of the coefficients:

- The coefficients of the expression are 7, -35, and 21.
- The greatest common factor of 7, 35, and 21 is 7.

2. Identify the GCF of the variable terms:

- The variable parts include [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] in these terms is [tex]\(x^2\)[/tex].

3. Combine the GCF of the coefficients and the variable terms:

- The overall greatest common factor of the expression is [tex]\(7x^2\)[/tex].

4. Factor out the GCF from each term:

- Divide each term by [tex]\(7x^2\)[/tex]:

- [tex]\(7x^4 \div 7x^2 = x^2\)[/tex]
- [tex]\(-35x^3 \div 7x^2 = -5x\)[/tex]
- [tex]\(21x^2 \div 7x^2 = 3\)[/tex]

5. Write the factored expression:

- The expression with the GCF factored out is:
[tex]\[
7x^4 - 35x^3 + 21x^2 = 7x^2(x^2 - 5x + 3)
\][/tex]

So, the correct choice is:

A. [tex]\(7x^4 - 35x^3 + 21x^2 = 7x^2(x^2 - 5x + 3)\)[/tex]