# Coefficients of an expansion

• Sep 9th 2009, 02:53 PM
Bingk
Coefficients of an expansion
Hello :)

I'm trying to prove that the coefficients $k_r$ for the expansion of $\prod_{i=1}^{n}(x-a_i)=x^n+k_{n-1}x^{n-1}+k_{n-2}x^{n-2}+\cdot\cdot\cdot+k_1x+(-1)^na_1a_2...a_n$ is

$k_s=(-1)^{n-s}\Bigl[\sum_{b_1

where $b_i \ \epsilon \ \{a_1,a_2,...,a_n\}$ and $0

Illustration:
For n = 4

$\prod_{i=1}^{4}(x-a_i)=$

$x^4-$
$(a_1+a_2+a_3+a_4)x^3+$
$(a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4)x^2-$
$(a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4)x+$
$a_1a_2a_3a_4$

(Sorry for the weird way of writing, but I had to chop it up cuz I get a latex error, in the preview, saying the image is too big).

Any help would be greatly appreciated :)