Equating coefficients

In mathematics, the method of equating the coefficients is a way of solving a functional equation of two polynomials for a number of unknown parameters. It relies on the fact that two polynomials are identical precisely when all corresponding coefficients are equal. The method is used to bring formulas into a desired form.

Example

Suppose we want to apply partial fraction decomposition to the equation::$frac\{1\}\{x(x-1)(x-2)\},,$that is, we want to bring it into the form::$frac\{a\}\{x\}+frac\{b\}\{x-1\}+frac\{c\}\{x-2\},,$in which the unknown parameters are "a", "b" and "c".Multiplying these formulas by $x(x-1)(x-2)$ turns both into polynomials, which we equate::$a(x-1)(x-2)\; +\; bx(x-2)\; +\; cx(x-1)\; =\; 1,,$or, after expansion and collecting terms with equal powers of $x$::$(a+b+c)x^2\; -\; (3a+2b+c)x\; +\; 2a\; =\; 1.,$At this point it is essential to realize that the polynomial $1$ is in fact equal to the polynomial $0x^2\; +\; 0x\; +\; 1$, having zero coefficients for the positive powers of $x$.Equating the corresponding coefficients now results in this system of linear equations::$a+b+c\; =\; 0,,$:$3a+2b+c\; =\; 0,,$:$2a\; =\; 1.,$Solving it results in::$a\; =\; frac\{1\}\{2\},,\; b\; =\; -1,,\; c\; =\; frac\{1\}\{2\}.,$