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 x^{2} 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 x^{3} are equal, and so with the coefficients of x^{2}, 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 (*):

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