To simplify the given expression [tex]\(\frac{5x^3 - 10x}{10x - 5x^2}\)[/tex], let’s go through the steps carefully.
1. Identify the numerator and the denominator:
- Numerator: [tex]\(5x^3 - 10x\)[/tex]
- Denominator: [tex]\(10x - 5x^2\)[/tex]
2. Factor both the numerator and the denominator:
- For the numerator [tex]\(5x^3 - 10x\)[/tex]:
[tex]\[
5x^3 - 10x = 5x(x^2 - 2)
\][/tex]
- For the denominator [tex]\(10x - 5x^2\)[/tex]:
[tex]\[
10x - 5x^2 = 5x(2 - x)
\][/tex]
So the expression now is:
[tex]\[
\frac{5x(x^2 - 2)}{5x(2 - x)}
\][/tex]
3. Cancel the common factors in the numerator and the denominator:
- Notice that both the numerator and the denominator have a common factor of [tex]\(5x\)[/tex]. We can cancel these common terms:
[tex]\[
\frac{5x(x^2 - 2)}{5x(2 - x)} = \frac{x^2 - 2}{2 - x}
\][/tex]
4. Simplify the remaining expression:
Now we have [tex]\(\frac{x^2 - 2}{2 - x}\)[/tex]. Let's manipulate the denominator to make it look similar to the numerator.
- Rewrite the denominator [tex]\(2 - x\)[/tex] as [tex]\(-(x - 2)\)[/tex]:
[tex]\[
2 - x = -(x - 2)
\][/tex]
Thus, the expression becomes:
[tex]\[
\frac{x^2 - 2}{-(x - 2)} = -\frac{x^2 - 2}{x - 2}
\][/tex]
5. Summarize the simplified form:
So, the simplified form of the expression is:
[tex]\[
-\frac{x^2 - 2}{x - 2}
\][/tex]
However, given our specific steps to factor and simplify, the expression most naturally simplifies directly to:
[tex]\[
\frac{2 - x^2}{x - 2}
\][/tex]
We can write this final simplified expression as:
[tex]\[
\frac{(2 - x^2)}{-(x - 2)} = -(2 - x^2)/(x - 2)
\][/tex]
To match the format of provided simplified expressions, one can reconfirm:
The fully simplified expression of [tex]\(\frac{5x^3 - 10x}{10x - 5x^2}\)[/tex] is:
[tex]\[
\frac{2 - x^2}{x - 2}
\][/tex]
