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Answer :
We are given the polynomial
[tex]$$
25x^4 + 15x^3 - 20x^2.
$$[/tex]
Step 1. Factor out the greatest common factor.
Notice that each term has a factor of [tex]$5x^2$[/tex], so we factor this out:
[tex]$$
25x^4 + 15x^3 - 20x^2 = 5x^2\left(\frac{25x^4}{5x^2} + \frac{15x^3}{5x^2} - \frac{20x^2}{5x^2}\right).
$$[/tex]
Simplify the expression inside the parentheses:
- [tex]$\displaystyle \frac{25x^4}{5x^2} = 5x^2$[/tex]
- [tex]$\displaystyle \frac{15x^3}{5x^2} = 3x$[/tex]
- [tex]$\displaystyle \frac{20x^2}{5x^2} = 4$[/tex]
So the expression becomes:
[tex]$$
5x^2\left(5x^2 + 3x - 4\right).
$$[/tex]
Step 2. Write the final answer.
The fully factored form of the polynomial is
[tex]$$
5x^2 \left(5x^2 + 3x - 4\right).
$$[/tex]
Thus, the answer is:
[tex]$$
25x^4 + 15x^3 - 20x^2 = 5x^2(5x^2 + 3x - 4).
$$[/tex]
[tex]$$
25x^4 + 15x^3 - 20x^2.
$$[/tex]
Step 1. Factor out the greatest common factor.
Notice that each term has a factor of [tex]$5x^2$[/tex], so we factor this out:
[tex]$$
25x^4 + 15x^3 - 20x^2 = 5x^2\left(\frac{25x^4}{5x^2} + \frac{15x^3}{5x^2} - \frac{20x^2}{5x^2}\right).
$$[/tex]
Simplify the expression inside the parentheses:
- [tex]$\displaystyle \frac{25x^4}{5x^2} = 5x^2$[/tex]
- [tex]$\displaystyle \frac{15x^3}{5x^2} = 3x$[/tex]
- [tex]$\displaystyle \frac{20x^2}{5x^2} = 4$[/tex]
So the expression becomes:
[tex]$$
5x^2\left(5x^2 + 3x - 4\right).
$$[/tex]
Step 2. Write the final answer.
The fully factored form of the polynomial is
[tex]$$
5x^2 \left(5x^2 + 3x - 4\right).
$$[/tex]
Thus, the answer is:
[tex]$$
25x^4 + 15x^3 - 20x^2 = 5x^2(5x^2 + 3x - 4).
$$[/tex]
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