Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(25x^5 + 35x^3 + 15x^2\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients are 25, 35, and 15.
- Find the GCF of these numbers:
- The prime factors of 25 are [tex]\(5 \times 5\)[/tex].
- The prime factors of 35 are [tex]\(5 \times 7\)[/tex].
- The prime factors of 15 are [tex]\(5 \times 3\)[/tex].
- The common factor among these numbers is 5, so the GCF of the coefficients is 5.
2. Identify the smallest power of [tex]\(x\)[/tex] present in all terms:
- The powers of [tex]\(x\)[/tex] in the polynomial are [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
3. Combine the GCF of the coefficients with the smallest power of [tex]\(x\)[/tex]:
- The overall GCF for the polynomial is [tex]\(5x^2\)[/tex].
4. Factor out the GCF from each term:
- For [tex]\(25x^5\)[/tex], factoring out [tex]\(5x^2\)[/tex] gives [tex]\(5x^2 \times 5x^3\)[/tex].
- For [tex]\(35x^3\)[/tex], factoring out [tex]\(5x^2\)[/tex] gives [tex]\(5x^2 \times 7x\)[/tex].
- For [tex]\(15x^2\)[/tex], factoring out [tex]\(5x^2\)[/tex] gives [tex]\(5x^2 \times 3\)[/tex].
5. Write the polynomial with the GCF factored out:
- The polynomial factored with the GCF is:
[tex]\[
5x^2(5x^3 + 7x + 3)
\][/tex]
So, the answer is that the polynomial [tex]\(25x^5 + 35x^3 + 15x^2\)[/tex] factored by the GCF is [tex]\(5x^2(5x^3 + 7x + 3)\)[/tex].
1. Identify the GCF of the coefficients:
- The coefficients are 25, 35, and 15.
- Find the GCF of these numbers:
- The prime factors of 25 are [tex]\(5 \times 5\)[/tex].
- The prime factors of 35 are [tex]\(5 \times 7\)[/tex].
- The prime factors of 15 are [tex]\(5 \times 3\)[/tex].
- The common factor among these numbers is 5, so the GCF of the coefficients is 5.
2. Identify the smallest power of [tex]\(x\)[/tex] present in all terms:
- The powers of [tex]\(x\)[/tex] in the polynomial are [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
3. Combine the GCF of the coefficients with the smallest power of [tex]\(x\)[/tex]:
- The overall GCF for the polynomial is [tex]\(5x^2\)[/tex].
4. Factor out the GCF from each term:
- For [tex]\(25x^5\)[/tex], factoring out [tex]\(5x^2\)[/tex] gives [tex]\(5x^2 \times 5x^3\)[/tex].
- For [tex]\(35x^3\)[/tex], factoring out [tex]\(5x^2\)[/tex] gives [tex]\(5x^2 \times 7x\)[/tex].
- For [tex]\(15x^2\)[/tex], factoring out [tex]\(5x^2\)[/tex] gives [tex]\(5x^2 \times 3\)[/tex].
5. Write the polynomial with the GCF factored out:
- The polynomial factored with the GCF is:
[tex]\[
5x^2(5x^3 + 7x + 3)
\][/tex]
So, the answer is that the polynomial [tex]\(25x^5 + 35x^3 + 15x^2\)[/tex] factored by the GCF is [tex]\(5x^2(5x^3 + 7x + 3)\)[/tex].
