3.4.9 Fraction Decomposition

Partial fraction decomposition is used to change a rational function of polynomials into a sum of simpler polynomials.

Partial Fraction provides the first step in a sequence of transformations by rewriting an expression of the form f(x)÷(g(x)⋅h(x)) as the equation f(x)÷(g(x)⋅h(x))=e_1(g(x))+e_2(h(x)) where the right side is the expansion of the denominator on the left side. Note the rewritten equation is useful in this form only if the factors in the denominator are irreducible polynomials.

In the simplest case, where g(x) and h(x) are polynomials of degree 1, the expansion is f(x)÷(g(x)⋅h(x))=A÷g(x)+B÷h(x). For example, (x+2)÷((x-1)⋅(x+1)) is rewritten as

In the general case, the denominator is a product of exponentiated polynomials and the number of terms in the expansion is the sum of the exponents. An exponentiated polynomial is simply a base polynomial raised to some power like (a_n⋅x^n+..+a_0⋅x^0)^m, which contributes m quotient terms. The numerator of each quotient term is an abstract polynomial^[1] of degree one less than the base polynomial. The denominator is the base polynomial raised to successively higher powers until m is reached. That is, the exponentiated polynomial (x^2+3)^3 contributes three terms of the form (A⋅x+B)÷(x^2+3)^n where n ranges from 1 to 3 and coefficients are generated so as to be unique to each quotient term. For example, the expression (x+1)÷((x+2)^2⋅(x^2+3)^3) expands to

Because Partial Fraction is just a first step, and because the subject expression is often part of a larger expression, the equation is introduced as a separate expression on the display. The new equation is then subjected to more manipulations before changes can be made to the original equation. See §4.4.4 for a complete example.

To continue with example (1) above, the denominator on the left of (1) is moved to the right side of the equation and simplified until an expression in terms of A, B, and x has been reached.

From this, A+B=1 and A-B=2, so A=1/2 and B=-3/2. The final result is