Equating coefficients
Encyclopedia
In mathematics, the method of equating the coefficients is a way of solving a functional equation of two polynomial
s 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.
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:
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s 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 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: