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Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(35x^5 + 5x^3 + 10x^2\)[/tex], follow these steps:
1. Identify the GCF:
- Look at the numerical coefficients: 35, 5, and 10. The GCF of these numbers is 5 since 5 is the largest number that divides all these coefficients without leaving a remainder.
- Look at the variable part: [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]. The lowest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]. Therefore, the GCF of the variable terms is [tex]\(x^2\)[/tex].
2. Combine the GCFs:
- The overall GCF of the entire polynomial is [tex]\(5x^2\)[/tex].
3. Factor the GCF out of the polynomial:
- Divide each term in the polynomial by the GCF:
- [tex]\(35x^5 \div 5x^2 = 7x^3\)[/tex]
- [tex]\(5x^3 \div 5x^2 = x\)[/tex]
- [tex]\(10x^2 \div 5x^2 = 2\)[/tex]
4. Write the factored expression:
- The polynomial [tex]\(35x^5 + 5x^3 + 10x^2\)[/tex] can be factored as:
[tex]\[
5x^2(7x^3 + x + 2)
\][/tex]
Therefore, the factored form by taking out the GCF is [tex]\(5x^2(7x^3 + x + 2)\)[/tex].
1. Identify the GCF:
- Look at the numerical coefficients: 35, 5, and 10. The GCF of these numbers is 5 since 5 is the largest number that divides all these coefficients without leaving a remainder.
- Look at the variable part: [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]. The lowest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]. Therefore, the GCF of the variable terms is [tex]\(x^2\)[/tex].
2. Combine the GCFs:
- The overall GCF of the entire polynomial is [tex]\(5x^2\)[/tex].
3. Factor the GCF out of the polynomial:
- Divide each term in the polynomial by the GCF:
- [tex]\(35x^5 \div 5x^2 = 7x^3\)[/tex]
- [tex]\(5x^3 \div 5x^2 = x\)[/tex]
- [tex]\(10x^2 \div 5x^2 = 2\)[/tex]
4. Write the factored expression:
- The polynomial [tex]\(35x^5 + 5x^3 + 10x^2\)[/tex] can be factored as:
[tex]\[
5x^2(7x^3 + x + 2)
\][/tex]
Therefore, the factored form by taking out the GCF is [tex]\(5x^2(7x^3 + x + 2)\)[/tex].
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