Equating coefficients

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}.,


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