#### 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 (*): #### Find the partial fraction decomposition.

1. 2. 3. 1. 2. 3. 