Answer :
To find the simplified form of the expression [tex]\(\frac{15 x^8 + 10 x^5 - 20 x^6 + 35 x^2}{5 x^3}\)[/tex], follow these steps:
1. Divide Each Term by [tex]\(5x^3\)[/tex]:
The expression can be broken down by dividing each term of the numerator by [tex]\(5x^3\)[/tex]:
[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 Fraction:
- [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 the simplified terms into a single expression:
[tex]\[
3x^5 + 2x^2 - 4x^3 + \frac{7}{x}
\][/tex]
4. Reorder the Terms:
To present it more clearly, reorder the terms in descending powers of [tex]\(x\)[/tex]:
[tex]\[
3x^5 - 4x^3 + 2x^2 + \frac{7}{x}
\][/tex]
This resulting expression is the simplified form of the original fraction.
1. Divide Each Term by [tex]\(5x^3\)[/tex]:
The expression can be broken down by dividing each term of the numerator by [tex]\(5x^3\)[/tex]:
[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 Fraction:
- [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 the simplified terms into a single expression:
[tex]\[
3x^5 + 2x^2 - 4x^3 + \frac{7}{x}
\][/tex]
4. Reorder the Terms:
To present it more clearly, reorder the terms in descending powers of [tex]\(x\)[/tex]:
[tex]\[
3x^5 - 4x^3 + 2x^2 + \frac{7}{x}
\][/tex]
This resulting expression is the simplified form of the original fraction.
