I think I have figured it out.

Given say three arbitrary functions of x, , and , the constants and the following equation:

Can be written:

If the functions are linearly independent then the only solution to the equation is when the coefficients of the functions are all zero. This leads to:

Which shows the coefficients of each side of the original equation are equated. The idea can be extended to any number of linearly independent functions.

The "x" terms of a polynomial are linearly independent so the coefficients can be equated also.