Answer :
To factor out the greatest common factor (GCF) from the expression [tex]\(5x^4 - 35x^3 + 15x^2\)[/tex], we'll follow these steps:
1. Identify the GCF of the coefficients:
Look at the coefficients of the terms: 5, 35, and 15.
- The GCF of these numbers is 5.
2. Look at the variables:
Each term shares the variable [tex]\(x\)[/tex]. Consider the smallest power of [tex]\(x\)[/tex] that appears in the expression:
- The powers are [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power is [tex]\(x^2\)[/tex].
3. Combine these two parts for the overall GCF:
- The GCF of the expression is the product of the GCF of the coefficients and the lowest power of [tex]\(x\)[/tex]: [tex]\(5x^2\)[/tex].
4. Factor the GCF out of each term:
- For [tex]\(5x^4\)[/tex], after factoring out [tex]\(5x^2\)[/tex], we get:
[tex]\[
5x^4 \div 5x^2 = x^2
\][/tex]
- For [tex]\(-35x^3\)[/tex], after factoring out [tex]\(5x^2\)[/tex], we get:
[tex]\[
-35x^3 \div 5x^2 = -7x
\][/tex]
- For [tex]\(15x^2\)[/tex], after factoring out [tex]\(5x^2\)[/tex], we get:
[tex]\[
15x^2 \div 5x^2 = 3
\][/tex]
5. Write the completely factored expression:
So, the expression [tex]\(5x^4 - 35x^3 + 15x^2\)[/tex] factored by the greatest common factor is:
[tex]\[
5x^2(x^2 - 7x + 3)
\][/tex]
This is the expression with the GCF factored out.
1. Identify the GCF of the coefficients:
Look at the coefficients of the terms: 5, 35, and 15.
- The GCF of these numbers is 5.
2. Look at the variables:
Each term shares the variable [tex]\(x\)[/tex]. Consider the smallest power of [tex]\(x\)[/tex] that appears in the expression:
- The powers are [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power is [tex]\(x^2\)[/tex].
3. Combine these two parts for the overall GCF:
- The GCF of the expression is the product of the GCF of the coefficients and the lowest power of [tex]\(x\)[/tex]: [tex]\(5x^2\)[/tex].
4. Factor the GCF out of each term:
- For [tex]\(5x^4\)[/tex], after factoring out [tex]\(5x^2\)[/tex], we get:
[tex]\[
5x^4 \div 5x^2 = x^2
\][/tex]
- For [tex]\(-35x^3\)[/tex], after factoring out [tex]\(5x^2\)[/tex], we get:
[tex]\[
-35x^3 \div 5x^2 = -7x
\][/tex]
- For [tex]\(15x^2\)[/tex], after factoring out [tex]\(5x^2\)[/tex], we get:
[tex]\[
15x^2 \div 5x^2 = 3
\][/tex]
5. Write the completely factored expression:
So, the expression [tex]\(5x^4 - 35x^3 + 15x^2\)[/tex] factored by the greatest common factor is:
[tex]\[
5x^2(x^2 - 7x + 3)
\][/tex]
This is the expression with the GCF factored out.
