Homework:
8.3a and 8.3b on Math XL
8.3: Partial Fraction Decomposition
Warm Up:
Simplify:
1
𝑥+1
+
2
𝑥
by finding a common denominator
Partial Fractions
• Partial fraction decomposition is the reverse of the previous
problem.
• i.e. How can we take an expression like
1
𝑥+1
2
3𝑥+2
𝑥 2 +𝑥
and turn it into
+ ?
𝑥
• In general, how can we turn a rational expression into the
sum/difference of multiple rational expressions?
More on Partial Fractions
• We can only do this with proper rational expressions (the degree
of the numerator < the degree of the denominator)
• Our strategy that we take depends on the denominator of the
original expression. There are 4 cases to consider:
1. Denominator has non-repeated (distinct) linear factors.
2. Denominator has repeated linear factors.
3. Denominator has a non-repeated irreducible (prime) quadratic
factor.
4. Denominator has a repeated irreducible quadratic factor.
Case 1: Denominator has distinct (non-repeated) linear factors
1. Write the partial fraction decomposition of each expression.
a.
2
𝑥(𝑥−2)
b.
5𝑥−3
𝑥 2 −2𝑥−3
c.
𝑥
𝑥 2 +2𝑥−3
Case 2: Denominator has repeated Linear Factors
d.
3𝑥 2 +14𝑥+12
𝑥(𝑥+2)2
e.
𝑥 2 −𝑥−8
(𝑥+3)(𝑥 2 +6𝑥+9)
Case 3: Denominator has non-repeated prime (irreducible) factors
f.
𝑥+4
𝑥 2 (𝑥 2 +4)
g.
2𝑥+4
𝑥 3 −1
Case 4: Denominator has a repeated irreducible quadratic factor
h.
𝑥 2 +2𝑥+1
(𝑥 2 +1)2
i.
𝑥 2 +2𝑥+3
(𝑥 2 +4)2
2. Set up the partial fraction decomposition for the expressions.
Don’t solve for A,B,C, etc., just set it up.
a.
b.
c.
5
𝑥 3 (𝑥−1)2
3𝑥
(𝑥 2 +4)(3𝑥+2)3
𝑥 2 +3𝑥
(2𝑥+5)2 (𝑥 2 +5𝑥+2)2