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Math Help - Simplifying expressions - algebra 2

  1. #1
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    Simplifying expressions - algebra 2

    1) (x^3 - x^2 - 4x + 4)/ (x^3 - 4x)?

    The answer is supposed to be x-1/x, how do I get there

    I simplify it and all I get to is x^3 - (x-2)/ x(x+2).....anyone give me some help????

    2) [1-h(k^-1)]/[ (h^-1) - (k^-1)]

    1 minus h times k to the -1 divided/over h to the -1 minus k to the -1

    The answer is supposed to be h..how do I get there? I'm clueless on this one. My fractions get all mixed up and stuff, I get confused on negative exponents...
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Dunit0001 View Post
    1) (x^3 - x^2 - 4x + 4)/ (x^3 - 4x)?

    The answer is supposed to be x-1/x, how do I get there

    I simplify it and all I get to is x^3 - (x-2)/ x(x+2).....anyone give me some help????

    2) [1-h(k^-1)]/[ (h^-1) - (k^-1)]

    1 minus h times k to the -1 divided/over h to the -1 minus k to the -1

    The answer is supposed to be h..how do I get there? I'm clueless on this one. My fractions get all mixed up and stuff, I get confused on negative exponents...
    1)
    \frac{x^3 - x^2 - 4x + 4}{x^3 - 4x} = \frac{(x^3 - 4x) - (x^2 - 4)}{x(x^2 - 4)}

    so, can you continue now?

    2)
    \frac{1 -hk^{-1}}{h^{-1} - k^{-1}} = \frac{1 - \frac{h}{k}}{\frac{1}{h} - \frac{1}{k}} = \frac{\frac{k - h}{k}}{\frac{k - h}{hk}}

    continue it..
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  3. #3
    Super Member angel.white's Avatar
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    Quote Originally Posted by Dunit0001 View Post
    1) (x^3 - x^2 - 4x + 4)/ (x^3 - 4x)?

    The answer is supposed to be x-1/x, how do I get there

    I simplify it and all I get to is x^3 - (x-2)/ x(x+2).....anyone give me some help????

    2) [1-h(k^-1)]/[ (h^-1) - (k^-1)]

    1 minus h times k to the -1 divided/over h to the -1 minus k to the -1

    The answer is supposed to be h..how do I get there? I'm clueless on this one. My fractions get all mixed up and stuff, I get confused on negative exponents...
    \frac{x^{3}-x^{2}-4x+4}{x^{3}-4x}

    First lets factor the denominator, you can see that both terms have a common factor of x, so lets factor that out.

    \frac{x^{3}-x^{2}-4x+4}{x(x^{2}-4)}

    Now, you should be able to see that we could further factor it to x(x-2)(x+2), but lets first see what we can do with it like it is.

    Do you see any common factors in the numerator? It's a bit subtle, but if you look for it you can see that x^{3}-x^{2} = x^{2}(x-1) and also -4x+4= -4(x-1)

    So you can rewrite the numerator as x^{3}-x^{2}-4x+4=x^{2}(x-1) -4(x-1)

    Now, each of these terms has a common factor of (x-1) so lets factor that out. x^{2}(x-1) -4(x-1) = (x-1)(x^{2}-4)

    Now lets replace our old unfactored numerator with our new factored numerator.

    \frac{x^{3}-x^{2}-4x+4}{x(x^{2}-4)}=\frac{(x-1)(x^{2}-4)}{x(x^{2}-4)}

    Now you can see that (x^{2}-4) is in both the numerator and denominator, so they cancel eachother out.
    \frac{(x-1)(x^{2}-4)}{x(x^{2}-4)}=\frac{(x-1)}{x}

    This can be split up as follows
    \frac{(x-1)}{x}=\frac{x}{x}-\frac{1}{x}

    Which simplifies to
    \frac{x}{x}-\frac{1}{x}=1-\frac{1}{x}
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