Answer :
To factor out the greatest common factor (GCF) from the polynomial [tex]\(35x^5 - 25x^2 + 40x\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients of the terms are 35, 25, and 40.
- The GCF of 35, 25, and 40 is 5. This is because 5 is the largest positive integer that divides all three numbers without leaving a remainder.
2. Identify the GCF of the variable parts:
- The variable parts of the terms are [tex]\(x^5\)[/tex], [tex]\(x^2\)[/tex], and [tex]\(x\)[/tex].
- The smallest power of [tex]\(x\)[/tex] among the terms is [tex]\(x\)[/tex]. So, the GCF for the variables is [tex]\(x\)[/tex].
3. Combine the GCF of the coefficients and the variable parts:
- The overall GCF of the polynomial is [tex]\(5x\)[/tex].
4. Factor out the GCF from each term in the polynomial:
- For the first term: [tex]\(35x^5\)[/tex]. Divide by the GCF ([tex]\(5x\)[/tex]):
[tex]\[
\frac{35x^5}{5x} = 7x^4
\][/tex]
- For the second term: [tex]\(-25x^2\)[/tex]. Divide by the GCF ([tex]\(5x\)[/tex]):
[tex]\[
\frac{-25x^2}{5x} = -5x
\][/tex]
- For the third term: [tex]\(40x\)[/tex]. Divide by the GCF ([tex]\(5x\)[/tex]):
[tex]\[
\frac{40x}{5x} = 8
\][/tex]
5. Write the factored form of the polynomial:
Combine the results from dividing each term by the GCF, and write the polynomial as:
[tex]\[
35x^5 - 25x^2 + 40x = 5x(7x^4 - 5x + 8)
\][/tex]
So, the factored form of the polynomial is [tex]\(5x(7x^4 - 5x + 8)\)[/tex].
1. Identify the GCF of the coefficients:
- The coefficients of the terms are 35, 25, and 40.
- The GCF of 35, 25, and 40 is 5. This is because 5 is the largest positive integer that divides all three numbers without leaving a remainder.
2. Identify the GCF of the variable parts:
- The variable parts of the terms are [tex]\(x^5\)[/tex], [tex]\(x^2\)[/tex], and [tex]\(x\)[/tex].
- The smallest power of [tex]\(x\)[/tex] among the terms is [tex]\(x\)[/tex]. So, the GCF for the variables is [tex]\(x\)[/tex].
3. Combine the GCF of the coefficients and the variable parts:
- The overall GCF of the polynomial is [tex]\(5x\)[/tex].
4. Factor out the GCF from each term in the polynomial:
- For the first term: [tex]\(35x^5\)[/tex]. Divide by the GCF ([tex]\(5x\)[/tex]):
[tex]\[
\frac{35x^5}{5x} = 7x^4
\][/tex]
- For the second term: [tex]\(-25x^2\)[/tex]. Divide by the GCF ([tex]\(5x\)[/tex]):
[tex]\[
\frac{-25x^2}{5x} = -5x
\][/tex]
- For the third term: [tex]\(40x\)[/tex]. Divide by the GCF ([tex]\(5x\)[/tex]):
[tex]\[
\frac{40x}{5x} = 8
\][/tex]
5. Write the factored form of the polynomial:
Combine the results from dividing each term by the GCF, and write the polynomial as:
[tex]\[
35x^5 - 25x^2 + 40x = 5x(7x^4 - 5x + 8)
\][/tex]
So, the factored form of the polynomial is [tex]\(5x(7x^4 - 5x + 8)\)[/tex].
