Answer :
To simplify the expression [tex]\(\frac{15x^8 + 10x^5 - 20x^6 + 35x^2}{5x^3}\)[/tex], we can follow these steps:
1. Divide each term in the numerator by the denominator:
The expression can be broken down term by term:
[tex]\[
\frac{15x^8}{5x^3} + \frac{10x^5}{5x^3} - \frac{20x^6}{5x^3} + \frac{35x^2}{5x^3}
\][/tex]
2. Simplify each term individually:
- [tex]\(\frac{15x^8}{5x^3} = 3x^{8-3} = 3x^5\)[/tex]
- [tex]\(\frac{10x^5}{5x^3} = 2x^{5-3} = 2x^2\)[/tex]
- [tex]\(\frac{20x^6}{5x^3} = 4x^{6-3} = 4x^3\)[/tex]
- [tex]\(\frac{35x^2}{5x^3} = 7x^{2-3} = \frac{7}{x}\)[/tex]
3. Combine the simplified terms:
Combine all the simplified terms together to get:
[tex]\[
3x^5 + 2x^2 - 4x^3 + \frac{7}{x}
\][/tex]
4. Reorder the terms:
Arrange these terms in descending order of the powers of [tex]\(x\)[/tex]:
[tex]\[
3x^5 - 4x^3 + 2x^2 + \frac{7}{x}
\][/tex]
The final simplified expression is:
[tex]\[
(3x^5 - 4x^3 + 2x^2 + \frac{7}{x})
\][/tex]
This expression matches the result provided.
1. Divide each term in the numerator by the denominator:
The expression can be broken down term by term:
[tex]\[
\frac{15x^8}{5x^3} + \frac{10x^5}{5x^3} - \frac{20x^6}{5x^3} + \frac{35x^2}{5x^3}
\][/tex]
2. Simplify each term individually:
- [tex]\(\frac{15x^8}{5x^3} = 3x^{8-3} = 3x^5\)[/tex]
- [tex]\(\frac{10x^5}{5x^3} = 2x^{5-3} = 2x^2\)[/tex]
- [tex]\(\frac{20x^6}{5x^3} = 4x^{6-3} = 4x^3\)[/tex]
- [tex]\(\frac{35x^2}{5x^3} = 7x^{2-3} = \frac{7}{x}\)[/tex]
3. Combine the simplified terms:
Combine all the simplified terms together to get:
[tex]\[
3x^5 + 2x^2 - 4x^3 + \frac{7}{x}
\][/tex]
4. Reorder the terms:
Arrange these terms in descending order of the powers of [tex]\(x\)[/tex]:
[tex]\[
3x^5 - 4x^3 + 2x^2 + \frac{7}{x}
\][/tex]
The final simplified expression is:
[tex]\[
(3x^5 - 4x^3 + 2x^2 + \frac{7}{x})
\][/tex]
This expression matches the result provided.
