# Math Help - Where does simplification stop?

1. ## Where does simplification stop?

Revising basic algebra, I came across the exercise: Simplify 5/(x + 1) + 2/(x+4).
No worries, I thought.
I wrote: 5(x+4)+2(x+1) / (x+1)(x+4) = 5x+20+2x+2 / x2+5x+4 = 7x+22 / x2+5x+ 4.
But the book answer was: 7x+22 / (x+1)(x+4)
which told me that I did right to multiply out the top part of the fraction and collect like terms, but I should have left the bottom half alone, even though
it can be correctly written as x2+5x+4.
Why is this? Is it because we want to preserve first order terms as long as possible? Or is there some other basic principle I've failed to grasp?

2. ## Re: Where does simplification stop?

If you can factor an expression then it's best to do so, as it makes the expression a bit more "use friendly." The expression (x+4)(x+1) tells you right away what the two roots are (-4 and -1), as opposed to x^2+5x+4. Also it makes it easier to see that there is no more simplification possible - suppose that for this problem the numerator had turned out to be 7x+28, which you can factor to 7(x+4); with the denominator in the form (x+4)(x+1) you can see right away that the (x+4) terms cancel, whereas leaving the denominator as x^2+5x+4 would make it less obvious.

3. ## Re: Where does simplification stop?

Thanks very much for that swift and clear reply (with its dark hint of the quadratics to come!). I suppose I was puzzled because I was encouraged to multiply out the expression on top but not the one at the bottom. I see that the difference is that + sign in the middle, which tells me that the terms on either side are not "factors".
It's been a long time.