Equating coefficients

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 denest the nested radicals ${\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}}$ to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of rational parameters d, e such that

Squaring both sides of this equation yields:

To find d and e we equate the terms not involving square roots, so ${\displaystyle a=d+e,}$ and equate the parts involving radicals, so ${\displaystyle b{\sqrt {c}}=2{\sqrt {de}}}$ which when squared implies ${\displaystyle b^{2}c=4de.}$ This gives us two equations, one quadratic and one linear, in the desired parameters d and e, and these can be solved to obtain

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