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Math Help - ((3x/(2x-1))+8) Is this a Polynomial

  1. #1
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    ((3x/(2x-1))+8) Is this a Polynomial

    Please tell me if ((3x/(2x-1))+8) is a Polynomial or not

    I guess you can't simplify it and it's not a polynomial as the denominator is not resolved
    and if we try to solve it using binomial series then it will assume a negative exponent and won't stay a polynomial as polynomials can't have negative powers of variables

    but some guys say it's a polynomial please explain with complete reason if its a polynomial or not

    Thanks in anticipation.
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  2. #2
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    Re: ((3x/(2x-1))+8) Is this a Polynomial

    Hey maddymath.

    Its a polynomial only if you can simplify it into linear combinations of positive whole number powers of x (including a constant).

    One way apart from the binomial series is through the Taylor series expansion. If there is no Taylor series expansion that gives a bound on the number of non-zero terms then it means the function is not a polynomial.
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  3. #3
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    Re: ((3x/(2x-1))+8) Is this a Polynomial

    Quote Originally Posted by maddymath View Post
    Please tell me if ((3x/(2x-1))+8) is a Polynomial or not

    I guess you can't simplify it and it's not a polynomial as the denominator is not resolved
    and if we try to solve it using binomial series then it will assume a negative exponent and won't stay a polynomial as polynomials can't have negative powers of variables

    but some guys say it's a polynomial please explain with complete reason if its a polynomial or not

    Thanks in anticipation.
    \displaystyle \begin{align*} \frac{3x}{2x - 1} + 8 &= -3x \left( \frac{1}{1 - 2x} \right) + 8 \\ &= -3x \sum_{ k = 0}^{\infty} \left[ \left( 2x \right) ^k \right] + 8 \textrm{ provided } |x| < \frac{1}{2}  \end{align*}

    It can be written as an infinite polynomial, but not a finite one.
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