Answer :
To simplify the expression [tex]\(\frac{5x^7 - 45x^6 + 10x^2}{-5x^5}\)[/tex], follow these steps:
1. Factor Out Common Terms:
Begin by observing the numerator [tex]\(5x^7 - 45x^6 + 10x^2\)[/tex]. Notice that each term in the numerator has a factor of [tex]\(5x^2\)[/tex]. We can factor this out:
[tex]\[
5x^7 - 45x^6 + 10x^2 = 5x^2(x^5 - 9x^4 + 2)
\][/tex]
2. Simplify the Fraction:
Now, we substitute the factored expression back into the fraction:
[tex]\[
\frac{5x^2(x^5 - 9x^4 + 2)}{-5x^5}
\][/tex]
Recognize that we can cancel the [tex]\(5x^2\)[/tex] in the numerator with part of the [tex]\(-5x^5\)[/tex] in the denominator. When you divide [tex]\(5x^2\)[/tex] by [tex]\(-5x^5\)[/tex], you get:
[tex]\[
\frac{5x^2}{-5x^5} = -\frac{1}{x^3}
\][/tex]
3. Apply the Simplification:
Multiply the remaining expression in the parentheses by [tex]\(-\frac{1}{x^3}\)[/tex]:
[tex]\[
\left(x^5 - 9x^4 + 2\right) \times \left(-\frac{1}{x^3}\right)
\][/tex]
Distribute [tex]\(-\frac{1}{x^3}\)[/tex] across each term:
[tex]\[
-\frac{x^5}{x^3} + \frac{9x^4}{x^3} - \frac{2}{x^3}
\][/tex]
Simplify each term:
- [tex]\(\frac{x^5}{x^3} = x^{5-3} = x^2\)[/tex]
- [tex]\(\frac{9x^4}{x^3} = 9x^{4-3} = 9x\)[/tex]
- Finally, [tex]\(-\frac{2}{x^3}\)[/tex] remains as it is.
So, putting it all together, the simplified expression is:
[tex]\[
-x^2 + 9x - \frac{2}{x^3}
\][/tex]
This is the simplified form of the original expression [tex]\(\frac{5 x^7-45 x^6+10 x^2}{-5 x^5}\)[/tex].
1. Factor Out Common Terms:
Begin by observing the numerator [tex]\(5x^7 - 45x^6 + 10x^2\)[/tex]. Notice that each term in the numerator has a factor of [tex]\(5x^2\)[/tex]. We can factor this out:
[tex]\[
5x^7 - 45x^6 + 10x^2 = 5x^2(x^5 - 9x^4 + 2)
\][/tex]
2. Simplify the Fraction:
Now, we substitute the factored expression back into the fraction:
[tex]\[
\frac{5x^2(x^5 - 9x^4 + 2)}{-5x^5}
\][/tex]
Recognize that we can cancel the [tex]\(5x^2\)[/tex] in the numerator with part of the [tex]\(-5x^5\)[/tex] in the denominator. When you divide [tex]\(5x^2\)[/tex] by [tex]\(-5x^5\)[/tex], you get:
[tex]\[
\frac{5x^2}{-5x^5} = -\frac{1}{x^3}
\][/tex]
3. Apply the Simplification:
Multiply the remaining expression in the parentheses by [tex]\(-\frac{1}{x^3}\)[/tex]:
[tex]\[
\left(x^5 - 9x^4 + 2\right) \times \left(-\frac{1}{x^3}\right)
\][/tex]
Distribute [tex]\(-\frac{1}{x^3}\)[/tex] across each term:
[tex]\[
-\frac{x^5}{x^3} + \frac{9x^4}{x^3} - \frac{2}{x^3}
\][/tex]
Simplify each term:
- [tex]\(\frac{x^5}{x^3} = x^{5-3} = x^2\)[/tex]
- [tex]\(\frac{9x^4}{x^3} = 9x^{4-3} = 9x\)[/tex]
- Finally, [tex]\(-\frac{2}{x^3}\)[/tex] remains as it is.
So, putting it all together, the simplified expression is:
[tex]\[
-x^2 + 9x - \frac{2}{x^3}
\][/tex]
This is the simplified form of the original expression [tex]\(\frac{5 x^7-45 x^6+10 x^2}{-5 x^5}\)[/tex].
