# Simplification problem

#### nnaiaia

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
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 $$\displaystyle \frac{5x^2+4x+1}{x^2} = \frac{5x^2}{x^2} + \frac{4x}{x^2} + \frac{1}{x^2}$$

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