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Answer :
To factor the expression [tex]\(15x^6 + 35x^4 + 25x^3\)[/tex] completely, we can follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, we'll look for the GCF among the coefficients and the terms involving [tex]\(x\)[/tex].
- The coefficients are 15, 35, and 25. The GCF of these numbers is 5.
- For the powers of [tex]\(x\)[/tex], each term contains at least [tex]\(x^3\)[/tex]. So, the GCF of the powers of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
The GCF of the entire expression is [tex]\(5x^3\)[/tex].
2. Factor out the GCF:
We will factor out [tex]\(5x^3\)[/tex] from each term in the expression:
- From [tex]\(15x^6\)[/tex], factoring out [tex]\(5x^3\)[/tex] leaves us with [tex]\(3x^3\)[/tex].
- From [tex]\(35x^4\)[/tex], factoring out [tex]\(5x^3\)[/tex] leaves us with [tex]\(7x\)[/tex].
- From [tex]\(25x^3\)[/tex], factoring out [tex]\(5x^3\)[/tex] leaves us with 5.
After factoring out the GCF, we have:
[tex]\[
15x^6 + 35x^4 + 25x^3 = 5x^3(3x^3 + 7x + 5)
\][/tex]
3. Ensure the remaining expression is completely factored:
Look at the expression inside the parentheses: [tex]\(3x^3 + 7x + 5\)[/tex].
- This expression is a polynomial of degree 3. It doesn't factor further over the integers easily, and no obvious factors can be extracted.
Therefore, the completely factored form of the original expression is:
[tex]\[
5x^3(3x^3 + 7x + 5)
\][/tex]
This is the fully factored expression.
1. Identify the Greatest Common Factor (GCF):
First, we'll look for the GCF among the coefficients and the terms involving [tex]\(x\)[/tex].
- The coefficients are 15, 35, and 25. The GCF of these numbers is 5.
- For the powers of [tex]\(x\)[/tex], each term contains at least [tex]\(x^3\)[/tex]. So, the GCF of the powers of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
The GCF of the entire expression is [tex]\(5x^3\)[/tex].
2. Factor out the GCF:
We will factor out [tex]\(5x^3\)[/tex] from each term in the expression:
- From [tex]\(15x^6\)[/tex], factoring out [tex]\(5x^3\)[/tex] leaves us with [tex]\(3x^3\)[/tex].
- From [tex]\(35x^4\)[/tex], factoring out [tex]\(5x^3\)[/tex] leaves us with [tex]\(7x\)[/tex].
- From [tex]\(25x^3\)[/tex], factoring out [tex]\(5x^3\)[/tex] leaves us with 5.
After factoring out the GCF, we have:
[tex]\[
15x^6 + 35x^4 + 25x^3 = 5x^3(3x^3 + 7x + 5)
\][/tex]
3. Ensure the remaining expression is completely factored:
Look at the expression inside the parentheses: [tex]\(3x^3 + 7x + 5\)[/tex].
- This expression is a polynomial of degree 3. It doesn't factor further over the integers easily, and no obvious factors can be extracted.
Therefore, the completely factored form of the original expression is:
[tex]\[
5x^3(3x^3 + 7x + 5)
\][/tex]
This is the fully factored expression.
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