College

Factor out the GCF from the polynomial:

[tex]\[ 35x^4 - 25x^2 + 40x \][/tex]

[tex]\[ 35x^4 - 25x^2 + 40x = \square \][/tex]

Answer :

To factor out the greatest common factor (GCF) from the polynomial [tex]\(35x^4 - 25x^2 + 40x\)[/tex], follow these steps:

1. Identify the GCF of the coefficients:
- Look at the numerical coefficients: 35, 25, and 40.
- Find the greatest common factor (GCF) of these numbers:
- The factors of 35 are: 1, 5, 7, 35
- The factors of 25 are: 1, 5, 25
- The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40
- The common factor is 5, which is the greatest one.

2. Identify the GCF of the variables:
- Look at the variable part: [tex]\(x^4\)[/tex], [tex]\(x^2\)[/tex], and [tex]\(x\)[/tex].
- Each term contains at least one [tex]\(x\)[/tex].
- So the GCF of the variable part is [tex]\(x\)[/tex].

3. Combine the GCF of the coefficients and the variables:
- The overall GCF of the polynomial is [tex]\(5x\)[/tex].

4. Factor out the GCF:
- Divide each term in the polynomial by the GCF [tex]\(5x\)[/tex]:
- [tex]\(35x^4 \div 5x = 7x^3\)[/tex]
- [tex]\(-25x^2 \div 5x = -5x\)[/tex]
- [tex]\(40x \div 5x = 8\)[/tex]

5. Write the factored form:
- The polynomial [tex]\(35x^4 - 25x^2 + 40x\)[/tex] is factored as:
[tex]\[
5x(7x^3 - 5x + 8)
\][/tex]

So, the factored form of the polynomial is [tex]\(5x(7x^3 - 5x + 8)\)[/tex].

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