Simplification problem

May 2010
1
0
I'm looking at an example in my book. I have the following equation:
\(\displaystyle (5x^2+4x+1)/(x^2)\)

My book manages to simplify it to:

\(\displaystyle 5+(4/x)+(1/x^2)\)

How did it do this? I see that x^2 is in the denominator, I understand that it would cancel out \(\displaystyle 5x^2\) and leave us with 5. But I assumed that after that, the equation would look like this:

\(\displaystyle (5+4x+1)/1\) which equals \(\displaystyle (5+4x+1)\)

Which is incorrect. But I don't see why...
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
I'm looking at an example in my book. I have the following equation:
\(\displaystyle (5x^2+4x+1)/(x^2)\)

My book manages to simplify it to:

\(\displaystyle 5+(4/x)+(1/x^2)\)

How did it do this? I see that x^2 is in the denominator, I understand that it would cancel out \(\displaystyle 5x^2\) and leave us with 5. But I assumed that after that, the equation would look like this:

\(\displaystyle (5+4x+1)/1\) which equals \(\displaystyle (5+4x+1)\)

Which is incorrect. But I don't see why...
It's because by definition of a fraction, it's EVERYTHING in the numerator being divided by the denominator.

So \(\displaystyle \frac{5x^2+4x+1}{x^2} = \frac{5x^2}{x^2} + \frac{4x}{x^2} + \frac{1}{x^2}\)

\(\displaystyle = 5 + \frac{4}{x} + \frac{1}{x^2}\).