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Math Help - Prove this polynomial is nonzero.

  1. #1
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    Prove this polynomial is nonzero.

    Okay, this is frustrating. I can't seem to prove this elementary result.

    Let \lambda_1,\cdots,\lambda_k\in\mathbb{C} be distinct, i.e. \lambda_i\neq\lambda_j for i\neq j. Also, let a_1,\cdots,a_k\in\mathbb{C}\setminus\{0\}, i.e. complex nonzero values.

    Prove that \sum_{i=1}^ka_i\prod_{\substack{j=1\\j\neq i}}^k(\lambda_j-X) is a nonzero polynomial in X.

    I thought about using induction, but I can't see how to make that work. Directly expanding the coefficients is a nightmare, and one which doesn't seem to go anywhere either. I thought maybe there might be some linear algebra theory which would work, but I don't know. Any help would be much appreciated. Thanks!
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  2. #2
    Senior Member
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    Re: Prove this polynomial is nonzero.

    Solved. Just let [tex]X=\lambda_1[tex].
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