Let [itex]n[/itex] be a positive integer, and define the function(adsbygoogle = window.adsbygoogle || []).push({});

[itex]f_n(x_1,x_2,\ldots, x_n) = \left(

\begin{array}{ccccc}

1 & 1 & 1 & \ldots & 1 \\

x_1 & x_2 & x_3 & \ldots & x_n \\

x_1^2 & x_2^2 & x_3^2 & \ldots & x_n^2 \\

\vdots & \vdots & \vdots & \ddots & \vdots \\

x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \ldots & x_n^{n-1}

\end{array}

\right)[/itex]

(b) By considering the first column expansion of the determinant, show that

[itex]f_n(x_1,x_2,\ldots,x_2) = g_0 + g_1x_1 + g_2x_1^2 + \cdots + g_{n-1}x_1^{n-1}[/itex], and, in particular, [itex]g_{n-1}=(-1)^{n-1}f_{n-1}(x_2,x_3,\ldots,x_n)[/itex]

(c) Show that [itex]f_n(x_1,x_2,\ldots,x_2)[/itex] has [itex]x_i-x_1[/itex] as a factor, for all values of [itex]i[/itex] from 2 to [itex]n[/itex].

I think I've sorted part (b) out. I don't have a clue about how to proceed in part (c).

Thanks.

James

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: How to calculate this determinate

**Physics Forums | Science Articles, Homework Help, Discussion**