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Math Help - Polynomial Reduction Help

  1. #1
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    Polynomial Reduction Help

    Reduce to lowest terms:
    \frac {x^3+x^2+x+1}{x^3+3x^3+3x+1}

    I don't see how this can be factored to be reduced. Thanks.
    The answer is \frac{x^2+1}{x^2+2x+1}
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  2. #2
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    Re: Polynomial Reduction Help

    Just to be sure, is that really  3x^3 for the second term in the denominator?
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  3. #3
    MHF Contributor MarkFL's Avatar
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    Re: Polynomial Reduction Help

    For the numerator:

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

    You could use the rational roots theorem, which states that if the given polynomial has a rational root, it will come from the list x=\pm1. We can see that x=+1 cannot work, hence we find:

    f(-1)=-1+1-1+1=0

    So, we know x+1 is a factor of f(x). Use of division finds:

    f(x)=(x+1)(x^2+1)

    Now, for the denominator, we should recognize the binomial coefficients arising from the cube of a binomial, i.e.:

    x^3+3x^2+3x+1=(x+1)^3 and so we may state:

    \frac{x^3+x^2+x+1}{x^3+3x^2+3x+1}=\frac{(x+1)(x^2+  1)}{(x+1)^3}=\frac{x^2+1}{(x+1)^2}=\frac{x^2+1}{x^  2+2x+1}
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  4. #4
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    Re: Polynomial Reduction Help

    grillage - you are correct it is 3x^2. Thanks for the help markFL2, I didn't know about the rational roots theorem, that will make working with polynomials a whole lot easier.
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