College

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 :

To factor out the greatest common factor (GCF) from the expression [tex]\(7x^4 - 35x^3 + 21x^2\)[/tex], follow these steps:

1. Identify the GCF of the coefficients:
- Look at the numerical coefficients: 7, 35, and 21.
- The greatest number that divides all these coefficients is 7.

2. Identify the GCF of the variables:
- Look at the terms with variables: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The lowest power of [tex]\(x\)[/tex] that is common in all terms is [tex]\(x^2\)[/tex].

3. Combine the GCFs:
- The overall greatest common factor for the entire 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 using the GCF:
- Place the GCF outside the parentheses and write the remaining terms inside.

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

Therefore, the expression factored with the greatest common factor is:
[tex]\[
7x^4 - 35x^3 + 21x^2 = 7x^2(x^2 - 5x + 3)
\][/tex]

This means the correct choice is option A.

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