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Math Help - Polynomials and rational functions

  1. #1
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    Polynomials and rational functions

    The question asks "State whether the function is a polynomial, a rational function (but not a polynomial), or neither."

    This is the function:

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

    It looks rational, but apparently the answer is 'neither'. Why is this the case? Thanks.
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  2. #2
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    Hi

    This is due to the exponent 3/2
    To get a rational function you must have only integer exponents
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  3. #3
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    Ahh, thanks!
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  4. #4
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    Oh, also, does a polynomial have to have to have integer exponents?
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  5. #5
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    Yes
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  6. #6
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    Some examples

    F(x) = x^3 - 3x^{3/2} + 2x is NOT a polynomial

    F(x) = x^3 - 3x^2 + 2x is a polynomial

    F(x) = \frac{x^3 - 3x^{3/2} + 2x}{x^2 - 1} is NOT a rational function

    F(x) = \frac{x^3 - 3x^2 + 2x}{x^2 - 1} is a rational function


    Now a more tricky example

    F(x) = \frac{x^3 - 3x^2 + 2x}{x - 1} is a rational function

    But since x^3 - 3x^2 + 2x is equal to 0 when x=1, you can factor out x-1

    x^3 - 3x^2 + 2x = (x-1)(x^2 -  2x)

    Therefore F(x) = \frac{x^3 - 3x^2 + 2x}{x - 1} = \frac{(x-1)(x^2 -  2x)}{x - 1} = x^2-2x which is a polynomial

    Actually \frac{x^3 - 3x^2 + 2x}{x - 1} and x^2-2x are not totally equal since the first one is defined for all real except 1 whereas the second one is defined for all real

    The 2 functions are equal except on x=1
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