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Answer :
To factor the expression [tex]\(25x^5 + 15x^3 + 6x^2\)[/tex] completely, follow these steps:
1. Identify the greatest common factor (GCF):
- Look at the coefficients: 25, 15, and 6. The GCF of these numbers is 1, but we are mainly interested in the variable part here.
- Examine the variable part: all terms have at least [tex]\(x^2\)[/tex] as a factor. Therefore, the GCF of the variable part is [tex]\(x^2\)[/tex].
2. Factor out the greatest common factor:
- Remove [tex]\(x^2\)[/tex] from each term:
[tex]\[
25x^5 + 15x^3 + 6x^2 = x^2(25x^3 + 15x + 6)
\][/tex]
3. Check the remaining expression to see if it can be factored further:
- The inside of the parentheses, [tex]\(25x^3 + 15x + 6\)[/tex], needs to be examined. We aim to see if it can be factored into simpler polynomials, but in this case, it doesn't factor neatly into further polynomials of lower degree using simple methods.
4. Final expression:
- As a result, the completely factored form of the expression is:
[tex]\[
x^2(25x^3 + 15x + 6)
\][/tex]
Now, your expression is fully factored.
1. Identify the greatest common factor (GCF):
- Look at the coefficients: 25, 15, and 6. The GCF of these numbers is 1, but we are mainly interested in the variable part here.
- Examine the variable part: all terms have at least [tex]\(x^2\)[/tex] as a factor. Therefore, the GCF of the variable part is [tex]\(x^2\)[/tex].
2. Factor out the greatest common factor:
- Remove [tex]\(x^2\)[/tex] from each term:
[tex]\[
25x^5 + 15x^3 + 6x^2 = x^2(25x^3 + 15x + 6)
\][/tex]
3. Check the remaining expression to see if it can be factored further:
- The inside of the parentheses, [tex]\(25x^3 + 15x + 6\)[/tex], needs to be examined. We aim to see if it can be factored into simpler polynomials, but in this case, it doesn't factor neatly into further polynomials of lower degree using simple methods.
4. Final expression:
- As a result, the completely factored form of the expression is:
[tex]\[
x^2(25x^3 + 15x + 6)
\][/tex]
Now, your expression is fully factored.
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