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Math Help - Where does simplification stop?

  1. #1
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    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?
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  2. #2
    MHF Contributor ebaines's Avatar
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    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.
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  3. #3
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    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.
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