7.4 Partial Fraction Decomposition

A rational expression P / Q is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. Otherwise the rational expression is termed improper.

CASE 1: Q has only non-repeated linear factors.

Under the assumption that Q has only non-repeated linear factors, the polynomial Q has the form

where none of the numbers ai are equal. In this case, the partial fraction decomposition of P / Q is of the form

where the numbers Ai are to be determined.

Write a partial fraction decomposition of

Solution of this system is A=3, B=2.

CASE 2: Q has repeated linear factors.

If the polynomial Q has a repeated factor, say, (x - a) n, n > 2 an integer, then, in the partial fraction decomposition of P / Q, we allow for the terms

Write the partial fraction decomposition of

From the last and the first we easily get A and B.

A=-3/4, B=3/4, C=7/2.

Need to solve:

CASE 3: Q contains a non-repeated irreducible quadratic factor.

If Q contains a non-repeated irreducible quadratic factor of the form ax2 + bx + c, then, in the partial fraction decomposition of P / Q, allow for the term

where the numbers A and B are to be determined.

Write a partial fraction decomposition of

=

=

Collect like terms:

Comparing right and left sides gives:

Adding (1) and (3) gives

2A+2B = 4

Adding (2) and (4) gives

2A-2B = -8

Combining these

4A = -4

(1)

(2)

(3)

(4)

Solution is A = -1, B = 3, C = 2, D = 1.

CASE 4: Q contains repeated irreducible quadratic factors.

If the polynomial Q contains a repeated irreducible quadratic fator (ax2 + bx + c)n, n > 2, n an integer, then, in the partial fraction decomposition of P / Q, allow for the terms

where the numbers Ai ,Bi are to be determined.

Write the partial fraction decomposition of:

Comparing left and right sides leads to: