Equating coefficients

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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 polynomial are identical precisely when all corresponding coefficients are equal. The method is used to bring formulas into a desired form.

[edit] Example

Suppose we want to apply partial fraction decomposition to the formula

$\frac{1}{x(x-1)(x-2)},\,$

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 $0 x 2 + 0 x + 1$ , having zero coefficients for the positive powers of $x$ . Equating the corresponding coefficients now results in this system of linear equations:

$a+b+c = 0,\,$