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Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(25x^5 + 35x^3 + 10x^2\)[/tex], follow these steps:
1. Identify the GCF of the coefficients: Look at the coefficients of the polynomial terms, which are 25, 35, and 10. Determine the greatest common factor of these numbers.
- Factors of 25: 1, 5, 25
- Factors of 35: 1, 5, 7, 35
- Factors of 10: 1, 2, 5, 10
The largest number that is a factor of 25, 35, and 10 is 5.
2. Factor out common variables: Next, examine the variable parts of each term. Each term contains the variable [tex]\(x\)[/tex] raised to a power: [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]. The term with the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]. Therefore, [tex]\(x^2\)[/tex] is the greatest common variable factor.
3. Combine the GCFs: Combine the numerical GCF and the variable GCF. Thus, the overall GCF of the polynomial is [tex]\(5x^2\)[/tex].
4. Factor out the GCF from each term: Divide each term of the polynomial by [tex]\(5x^2\)[/tex] and write the resulting expression.
- First term: [tex]\(\frac{25x^5}{5x^2} = 5x^3\)[/tex]
- Second term: [tex]\(\frac{35x^3}{5x^2} = 7x\)[/tex]
- Third term: [tex]\(\frac{10x^2}{5x^2} = 2\)[/tex]
5. Write the factored form: Combine these results to express the polynomial as a product of its GCF and the remaining factors:
[tex]\[
25x^5 + 35x^3 + 10x^2 = 5x^2(5x^3 + 7x + 2)
\][/tex]
So, the fully factored form of the polynomial is [tex]\(5x^2(5x^3 + 7x + 2)\)[/tex].
1. Identify the GCF of the coefficients: Look at the coefficients of the polynomial terms, which are 25, 35, and 10. Determine the greatest common factor of these numbers.
- Factors of 25: 1, 5, 25
- Factors of 35: 1, 5, 7, 35
- Factors of 10: 1, 2, 5, 10
The largest number that is a factor of 25, 35, and 10 is 5.
2. Factor out common variables: Next, examine the variable parts of each term. Each term contains the variable [tex]\(x\)[/tex] raised to a power: [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]. The term with the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]. Therefore, [tex]\(x^2\)[/tex] is the greatest common variable factor.
3. Combine the GCFs: Combine the numerical GCF and the variable GCF. Thus, the overall GCF of the polynomial is [tex]\(5x^2\)[/tex].
4. Factor out the GCF from each term: Divide each term of the polynomial by [tex]\(5x^2\)[/tex] and write the resulting expression.
- First term: [tex]\(\frac{25x^5}{5x^2} = 5x^3\)[/tex]
- Second term: [tex]\(\frac{35x^3}{5x^2} = 7x\)[/tex]
- Third term: [tex]\(\frac{10x^2}{5x^2} = 2\)[/tex]
5. Write the factored form: Combine these results to express the polynomial as a product of its GCF and the remaining factors:
[tex]\[
25x^5 + 35x^3 + 10x^2 = 5x^2(5x^3 + 7x + 2)
\][/tex]
So, the fully factored form of the polynomial is [tex]\(5x^2(5x^3 + 7x + 2)\)[/tex].
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