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Math Help - Polynomial Shift

  1. #1
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    Polynomial Shift

    Suppose P(x) = anxn + an-1xn-1+an-2xn-2++a1x1+a0 is any polynomial of degree 1 or higher with real coefficients. Show that there must exist a real number k so that if we create the new polynomial Q(x) = P(x-k), then Q(x) = bnxn + bn-2xn-2 + + b1x1 + b0 meaning that the shift has caused the new polynomial to lose its x^(n-1) term.

    I know it has something to do with the significance of the 2nd coefficient polynomial, but I get lost after that. any help appreciated.
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  2. #2
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    Quote Originally Posted by amma0913 View Post

    Suppose P(x) = a_nx^n + a_{n-1}x^{n-1}+a_{n-2}x^{n-2} + \cdots +a_1x + a_0 is any polynomial of degree 1 or higher with real coefficients. Show that there must exist a real number k so that if we

    create the new polynomial Q(x) = P(x-k), then Q(x) = b_nx^n + b_{n-2}x^{n-2} + \cdots + b_1x + b_0 meaning that the shift has caused the new polynomial to lose its x^{n-1} term.

    I know it has something to do with the significance of the 2nd coefficient polynomial, but I get lost after that. any help appreciated.
    Hint: k=\frac{a_{n-1}}{na_n}.
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