Perform operations and simplify

[Kyle Concannon]

Please help with these rational problems

1.
_3_ _ _5__ + _2x______ Note: _____ is a straight line standing for division
x x^2 x^2+2x+1[/u][/code]

 

[stapel]

I'm sorry, but your fractions (I think) aren't lining up, at least not in my browser.

Please reply using single-line formatting, as explained in the "Karl's Notes" link in the "Forum Help" pull-down menu at the very top of the page. For instance, "two times x, divided by the sum of x squared, four times x, and four" would be written as:

. . . . .(2x) / (x^2 + 4x + 4)

Thank you.

Eliz.

 

[soroban]

Hello, Kyle!

\(\displaystyle 1)\;\;\L\frac{3}{x}\,-\,\frac{5}{x^2}\,+\,\frac{2x}{x^2\,+\,2x\,+\,1}\)

We have: .\(\displaystyle \frac{3}{x}\,-\,\frac{5}{x^2}\,+\,\frac{2x}{(x\,+\,1)^2}\)

The LCD is: \(\displaystyle x^2(x\,+\,1)^2\)

Convert the fractions to the LCD:

. . . \(\displaystyle \L\frac{3}{x}\,\cdot\,\frac{x(x\,+\,1)^2}{x(x\;+\;1)^2}\,-\,\frac{5}{x^2}\,\cdot\,\frac{(x\,+\,1)^2}{(x\,+\,1)^2}\;+\;\frac{2x}{(x\,+\,1)^2}\,\cdot\,\frac{x^2}{x^2}\)

. . \(\displaystyle \L=\;\frac{3x(x\,+\,1)^2 \,-\,5(x\,+\,1)^2\,+\,2x\cdot x^2}{x^2(x\,+\,1)^2}\)

. . \(\displaystyle \L=\:\frac{3x(x^2\,+\,2x\,+\,1)\,-\,5(x^2\,+\,2x\,+\,1)\,+\,2x^3}{x^2(x\,+\,1)^2}\)

. . \(\displaystyle \L=\;\frac{3x^3\,+\,6x^2\,+\,3x\,-\,5x^2\,-\,10x\,-\,5\,+\,2x^3}{x^2(x\,+\,1)^2}\)

. . \(\displaystyle \L=\;\frac{5x^3\,+\,x^2\,-\,7x\,-\,5}{x^2(x\,+\,1)^2}\)