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Math Help - (a^2 -1) / (-a -1) yet another simple one

  1. #1
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    (a^2 -1) / (-a -1) yet another simple one

    Simplify:

    (a^2 - 1) / (-a - 1)

    Answer key: 1 - a

    This looks like another simple problem. But I tried AfterShock method of splitting the numerator and find nothing to cancel; I also tried Dan's suggestion of factoring, I don't think I see something to factor. This one seems harder, how to arrive at 1 - a solution?

    Thanks.
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  2. #2
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    Quote Originally Posted by shenton View Post
    Simplify:

    (a^2 - 1) / (-a - 1)

    Answer key: 1 - a

    This looks like another simple problem. But I tried AfterShock method of splitting the numerator and find nothing to cancel; I also tried Dan's suggestion of factoring, I don't think I see something to factor. This one seems harder, how to arrive at 1 - a solution?

    Thanks.
    \frac{a^2 - 1}{-a - 1} = \frac{a^2 - 1}{-1(a + 1)} = -\frac{a^2 - 1}{a + 1}

    Now note that a^2 - 1 is the difference between two squares, so a^2 - 1 = (a + 1)(a - 1) so
    \frac{a^2 - 1}{-a - 1} = - \frac{(a + 1)(a - 1)}{(a + 1)} = -(a - 1) = 1 - a

    -Dan
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  3. #3
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    Quote Originally Posted by shenton View Post
    Simplify:

    (a^2 - 1) / (-a - 1)

    Answer key: 1 - a

    This looks like another simple problem. But I tried AfterShock method of splitting the numerator and find nothing to cancel; I also tried Dan's suggestion of factoring, I don't think I see something to factor. This one seems harder, how to arrive at 1 - a solution?

    Thanks.
    Another way, although the way shown is the easiest:

    (a^2 - 1)/(-a - 1) = (a^2)/(-a - 1) + (-1/(-a-1))

    = (-a^2)/(a + 1) + 1/(a + 1)

    In the above step, I just made it so that the denominator was positive and the numerator negative, and in the second both positive. Personal preference.

    We can expand the first set of terms by long division:

    (-a + 1) + -1/(a+1) + 1/(a+1) = (-a + 1)
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  4. #4
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    Thanks, guys for teaching. These questions are actually pretty hard. I learnt something now.
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