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Math Help - Can you simplify this further?

  1. #1
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    Can you simplify this further?

    Hi all,

    Had an exam recently and I've got a nagging feeling about a particular question.

    f(x) = \frac{1-x^2}{1-x} and g(t) = t^2

    Find f(g(t)) and simplify your answer.

    So that results in

    \frac{1-(t^2)^2}{1-t^2}

    Which simplifies to

    \frac{1-t^4}{1-t^2}

    Is there any further simplification we can do? I can't see anything but I have this nagging feeling I'm missing something. Possibly further factorisation taking place considering 1 is also 1^2 etc leading to some cancelling out.

    Did I miss anything?
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  2. #2
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    Yes, of course that simplifies. There is a general formula that a^n- b^n= (a-b)(a^{n-1}+ a^{n-2}b+ \cdot\cdot\cdot+ ab^{n-2}+ b^{n-1}. In particular
    1- t^2= (1- t)(1+ t)
    and
    1- t^4= (1- t)(1+ t+ t^2+ t^3)
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Yes, of course that simplifies. There is a general formula that a^n- b^n= (a-b)(a^{n-1}+ a^{n-2}b+ \cdot\cdot\cdot+ ab^{n-2}+ b^{n-1}. In particular
    1- t^2= (1- t)(1+ t)
    and
    1- t^4= (1- t)(1+ t+ t^2+ t^3)
    True, however I personally wouldn't consider

    \frac{1+ t+ t^2+ t^3}{1+ t}

    which that cancels down as a "simpler" answer than the original though, would you?
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  4. #4
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    Quote Originally Posted by Peleus View Post
    Hi all,

    Had an exam recently and I've got a nagging feeling about a particular question.

    f(x) = \frac{1-x^2}{1-x} and g(t) = t^2

    Find f(g(t)) and simplify your answer.

    So that results in

    \frac{1-(t^2)^2}{1-t^2}

    Which simplifies to

    \frac{1-t^4}{1-t^2}

    Is there any further simplification we can do? I can't see anything but I have this nagging feeling I'm missing something. Possibly further factorisation taking place considering 1 is also 1^2 etc leading to some cancelling out.

    Did I miss anything?
    Yes.

    \frac{1-t^4}{1-t^2} = \frac{(1 - t^2)(1 + t^2)}{1 - t^2} = 1 + t^2, t \neq \pm 1.
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  5. #5
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    Bother, knew I missed something.

    I originally had it as

    = \frac{(1-t^2)^2}{1-t^2}

    = \frac{(1-t^2)(1-t^2)}{1-t^2}

    = (1-t^2)

    But realised I wasn't correct because it was never squared, it was 1-(g(t))^2 so I scribbled it out and didn't spot the other way.
    Last edited by mr fantastic; April 23rd 2009 at 01:58 PM. Reason: Replaced a bug with the other
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