PARTIAL FRACTION DECOMPOSITION – QUADRATIC FACTORS

Partial fraction decomposition can be done for rational expressions of the form P(x) / Q(x) where P(x) and Q(x) have
no common factors and the degree of P(x) is less than the degree of Q(x). Partial fractions decomposition means
expressing P(x) / Q(x) as a sum of fractions. The denominator Q(x) determines how the partial fraction
decomposition will be done.

There are four cases:

           1. Q(x) is a product of distinct linear factors.

           2. Q(x) is a product of linear factors, some of which are repeated.

           3. Q(x) has distinct prime quadratic factors.

           4. Q(x) has a repeated prime quadratic factor.

This lesson covers cases 3 and 4.

Example 1

Since the fractions at the left and right sides have the same denominator, it follows that they have equal numerators.
Therefore, the coefficients of x² are equal, and so with the coefficients of x and the constant term.

A+B=1            (equation 1)

-A+C=0          (equation 2)

2A=10            (equation 3)

Solve this system of equations.

From equation 3: A=5

Substituting in equations 1 and 2: B= -4 , C=5

Therefore, substituting A, B, and C in (*):

Example 2

Since the fractions at the left and right sides have the same denominator, it follows that they have equal numerators.
Therefore, the coefficients of x³ are equal, and so with the coefficients of x², x, and the constant terms.

A=1                  (equation 1)

B= -2                (equation 2)

A+C=0              (equation 3)

|B+D=5            (equation 4)

Solve this system of equations.

Using A=1 in equation 3: C= -1

Using B= -2 in equation 4: D=7

Therefore, substituting A, B, C, and D in (*):

Try these problems

QUESTIONS

Find the partial fraction decomposition.

Answers