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Reverse order of polynomial coefficients of type $left(r-xright)^n$

Mathematics Asked by thinkingeye on January 18, 2021

I have a simple question.

I have a polynomial defined by
$$
left(r-xright)^n = a_0 + a_1 x + … + a_n x^n
$$

Is there a simple expression if I put the coefficients in reverse order?

$$
a_n + a_{n-1} x + … + a_1 x^{n-1} a_0 x^n = ?
$$

If $r = 1$ it is easy but for other $r$ I don’t know how to do it.

Edit solution:
Thanks to the general answer of paul blart math cop, the simple expression with reversed coefficient order is:

$$
x^n left(r-frac{1}{x}right)^n = left(rx-1right)^n
$$

One Answer

Letting $f(x)$ be a polynomial of degree $n$, the polynomial with reversed coefficients to $f$ is $x^n f(1/x)$.

Correct answer by paul blart math cop on January 18, 2021

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