In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters.
It relies on the fact that two expressions are identical precisely when corresponding coefficients are equal for each different type of term.
The method is used to bring formulas into a desired form.
Suppose we want to apply partial fraction decomposition to the expression: that is, we want to bring it into the form: 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: or, after expansion and collecting terms with equal powers of x: At this point it is essential to realize that the polynomial 1 is in fact equal to the polynomial 0x2 + 0x + 1, having zero coefficients for the positive powers of x. Equating the corresponding coefficients now results in this system of linear equations: Solving it results in: A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the nested radicals
This gives us two equations, one quadratic and one linear, in the desired parameters d and e, and these can be solved to obtain which is a valid solution pair if and only if
The method of equating coefficients is often used when dealing with complex numbers.
We write and multiply both sides by the denominator to obtain Equating real terms gives and equating coefficients of the imaginary unit i gives These are two equations in the unknown parameters e and f, and they can be solved to obtain the desired coefficients of the quotient: