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I have a question about when determining the weights in Gaussian quadrature for integrating a polynomial of degree 3: $c_0 + c_1x + c_2x^2 + c_3x^3$ up to the point when: \begin{align*} &c_0(\omega_0 + \omega_1 - \int_{-1}^1dx) + c_1(\omega_0x_0 + \omega_1x1 -\int_{-1}^1xdx) \\ &+ c_2(\omega_0x_0^2 + \omega_1x_1^2 - \int_{-1}^1x^2dx) + c_3(\omega_0x_0^3 + \omega_1x_1^3 - \int_{-1}^1x^3dx) = 0. \end{align*} I do not understand the explanation in the following:

Since $c_0,c_1,c_2,c_3$ are arbitrary, then their coefficients must all be 0.

Is this as saying: for $ax_1 + bx_2 + cx_3 + dx_4 = 0$, since $x_1, x_2, x_3, x_4$ are arbitrary, therefore a,b,c,d are 0?

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    $\begingroup$ If $ax_1 + bx_2 + cx_3 + dx_4 = 0$ for any choice of $x_1,x_2,x_3,x_4$, then $a=b=c=d=0$. To see why, simply let $x_1=1$ and $x_2=x_3=x_4=0$, thus $a=0$, etc. $\endgroup$ Commented Jan 2, 2014 at 22:52
  • $\begingroup$ Thank you arbautjc. Now I feel this is a very primitive question. LOL =] $\endgroup$ Commented Jan 2, 2014 at 22:57

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