Partial Fraction Decomposition (Summary)
Partial Fraction Decomposition is used when we have a fraction, P(x)/Q(x), where P, Q are
polynomials, and the degree of Pis less than the degree of Q.NOTE: If the degree of the
numerator is larger than the denominator, then perform long division first.
Assume Qis fully factored. We have 4 cases that we will consider.
Case I :Qhas distinct linear factors,
Q(x) = (a1x+b1)(a2x+b2). . . (akx+bk)
Then: P(x)
Q(x)=A1
a1x+b1
+A2
a2x+b2
+. . . +Ak
akx+bk
Example (be sure you could have found the constants):
3x
(x−1)(x−2) =A
x−2+B
x−1=6
x−2
−
3
x−1
Case II :Qhas some repeated linear factors. Let a1x+b1be repeated rtimes. Then, instead
of the single term A1/(a1x+b1), we have one term for each successive power in the
denominator: B1
a1x+b1
+B2
(a1x+b1)2+. . . +Br
(a1x+b1)r
Example:
3x+ 1
(x−1)2(x−2) =A
x−1+B
(x−1)2+C
x−2=−
7
x−1
−
4
(x−1)2+7
x−2
Case III :Qhas some irreducible quadratic factors, not repeated. Let ax2+bx +cbe an
irreducible quadratic factor for Q. Then the decomposition will have the term:
Ax +B
ax2+bx +c
Example: 3x
(x−1)(x2+ 1) =A
x−1+Bx +C
x2+ 1 =3
2
1
x−1+1
2
−3x+ 3
x2+ 1
Case IV :Qhas some irreducible quadratic factors, some repeated. Suppose that ax2+bx +cis
a repeated quadratic factor (repeated rtimes). Then, instead of the single expression
in Case III, we will have:
A1x+B1
ax2+bx +c+A2x+B2
(ax2+bx +c)2+. . . +Arx+Br
(ax2+bx +c)r
Example: x+ 3
(x−1)(x2+ 1)2=1
x−1
−
x+ 1
x2+ 1
−
2x+ 1
(x2+ 1)2
1
