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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]
### 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]
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