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2. Rational functions and partial fractions
2.1. Rational functions
A rational function is a function of the form
f (x) =
p(x) q(x)
where p(x) and q(x) are polynomials in x with q ≡ 0. For example
x + 3 x − 7
x − 2 2 x^3 + x^2 − x
x^2 + 3x + 2 1
The last is the same as x^2 + 3x + 2 , so any polynomial is also a rational function. If the numerator and denominator have a common factor, we can simplify the fraction by dividing top and bottom by that factor. For example,
x^2 + 3x + 2 x^2 + 2x + 1
(x + 1)(x + 2) (x + 1)^2
x + 2 x + 1
To multiply two rational functions, their numerators are multiplied together and their denominators are multiplied together. To divide two rational functions, turn the second one upside-down and multiply. For example, ( 4(x + 7) x + 1
÷
x^2 + 5 2 x + 2
4(x + 7) x + 1
×
2 x + 2 x^2 + 5
4(x + 7)(2x + 2) (x + 1)(x^2 + 5)
8(x + 7) x^2 + 5
To add or subtract two rational functions, you must write them using a common denom- inator. For example,
1 x + 1
x + 2
x + 2 (x + 1)(x + 2)
2(x + 1) (x + 1)(x + 2) = x + 2 + 2(x + 1) (x + 1)(x + 2)
3 x + 4 (x + 1)(x + 2)
Note. Often the common denominator is the product of the denominators, but sometimes you can take something smaller. For example,
2 x + 1
x (x + 1)(x + 2)
2(x + 2) (x + 1)(x + 2)
x (x + 1)(x + 2)
=
2(x + 2) − x (x + 1)(x + 2)
x + 4 (x + 1)(x + 2)
2.2. Proper rational functions
A proper rational function is one in which the degree of the numerator is less than the degree of the denominator. Otherwise it is called improper.
Any rational function can be written as the sum of a polynomial and a proper rational function.
Proof. Recall that if you divide a polynomial by a divisor, then
polynomial = divisor · quotient + remainder
Therefore polynomial divisor = quotient + remainder divisor
For example (using any earlier polynomial division),
2 x^3 + 10x^2 − 3 x + 1 x + 3 = (2x^2 + 4x − 15) +
x + 3
2.3. Partial fractions
The equality 3 x + 4 (x + 1)(x + 2)
x + 1
x + 2 expresses a complicated rational function as a sum of simple ones, a partial fraction. This is often useful.
Example 2.1. Consider 3 x − 1 (x + 1)(x − 3)
We try to write it as 3 x − 1 (x + 1)(x − 3)
A
x + 1
B
x − 3 where A and B are constants. Multiplying both sides by (x + 1)(x − 3) gives
3 x − 1 = A(x − 3) + B(x + 1).
This is an identity which is true for all x. Putting x = 3, it gives 8 = 4B, so B = 2. Putting x = − 1 , it gives −4 = − 4 A, so A = 1. With these values of A and B the identity does hold, for
3 x − 1 = (x − 3) + 2(x + 1).
Therefore 3 x − 1 (x + 1)(x − 3)
x + 1
x − 3
Since this is true for all x, we can compare coe cients. Therefore
x^2 : 0 = 3 + B x : 5 = 3 + C − B constant terms : 7 = 6 − C
Therefore B = − 3 and C = − 1 , and
5 x + 7 (x − 1)(x^2 + x + 2)
x − 1
3 x + 1 x^2 + x + 2
2.5. Repeated factors
So far each factor has occurred just once. If the denominator includes a factor like (x − a)^2 , we include partial fractions of the form
A x − a
B
(x − a)^2
Example 2.4. Write 3 x + 5 (x − 2)^2 in partial fractions.
Write 3 x + 5 (x − 2)^2
A
x − 2
B
(x − 2)^2
As usual, multiply the denominator to get
3 x + 5 = A(x − 2) + B.
Comparing coe cients gives A = 3 and B = 11, and so 3 x + 5 (x − 2)^2
x − 2
(x − 2)^2
Example 2.5. Write x^2 − 17 x − 8 (x − 3)(x + 2)^2 in partial fractions.
Write x^2 − 17 x − 8 (x − 3)(x + 2)^2
A
x − 3
B
x + 2
C
(x + 2)^2
Multiply the denominator to get
x^2 − 17 x − 8 =A(x + 2)^2 + B(x − 3)(x + 2) + C(x − 3) =A(x^2 + 4x + 4) + B(x^2 − x + 6) + C(x − 3).
Put x = 3 to get 9 − 17 × 3 − 8 = 25A, so A = − 2. Put x = − 2 to get 4 + 17 × 2 − 8 = − 5 C, so C = − 6. Compare coe cients of x^2 to get 1 = A + B, so B = 3.
Therefore x^2 − 17 x − 8 (x − 3)(x + 2)^2
x − 3
x + 2
(x + 2)^2
2.6. Worked examples
Example 2.6. Simplify the following as much as possible:
(x + 3)^2 (x^2 + 1)(x + 3)
(x^2 − 2 x − 8) (x + 2)
Example 2.7. Express the following as a single fraction, simplifying as much as possible: 1 (x − 2)
(x − 3)
(x + 2) (x − 5)
(x − 4) (x + 2)
(x − 1)
Example 2.8. Write the following as the sum of a polynomial and a proper rational function: (x^3 + 4x − 3) (x − 2)
(x^2 − 1) (x^2 + 1)
Example 2.9. Write as partial fractions: 2 (x + 1)(x − 1)
(2x + 1) x(x + 1)
(3 + 2x − x^2 ) (x − 1)(2x^2 + x + 1)
(3x^2 + x + 4) (x + 1)^2 (x − 1)
(2x^2 + 2x − 17) (x + 3)(x + 2)
