College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Factor the greatest common factor (GCF) out of the polynomial below:

[tex]\[ 25x^5 + 35x^3 + 10x^2 \][/tex]

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].