PARTIAL FRACTIONS

1. Introduction

Consider this addition.

The reverse process of starting withand writing it asis known as writing the expression in partial fractions.

This process is useful in integration, differentiation, and the binomial theorem (later topics in C4).

C4 p2 Ex 1A (if you feel practice is needed in addition and subtraction of algebraic fractions)

There are three types of partial fractions we are concerned with in C4, shown in the table below.

denominator type / example / partial fractions
two linear factors / /
three linear factors / /
repeated linear factor / /

Activity1:Probably for Further Mathematicians only! Discuss why the partial fractions have these forms. In particular, why is it necessary to have the second AND the third fractions for the repeated linear factor case?

2. Two Linear Factors

Example 1 : Write the following in partial fractions.

Suppose

Comparing numerators,

We now choose cunning values of x.

Therefore we have

Example 2 : Write the following in partial fractions.

Suppose

Comparing numerators,

Therefore we have

C4 p3 Ex 1B

3. Three Linear Factors

Example 1 : Write the following in partial fractions.

We begin by factorising the denominator as far as possible.

We have three linear factors. So,

Comparing numerators (once the RHS has been written as a single fraction),

Therefore

C4 p4 Ex 1C

4. Repeated Linear Factor

Example 1 : Write the following in partial fractions.

Suppose

Comparing numerators,

We cannot get B by making brackets worth zero. However, there are two alternative methods…

method 1 : pick any value of x / method 2 : comparing coefficients
/ /
/ Equating coefficients of on either side of the identity, (we could use coefficients of x or constant terms)

Example2:Write the following in partial fractions.

Suppose

Comparing numerators,

Comparing coefficients of ,

Comparing constant terms (we could equally well compare coefficients of x)

And so

C4 p5 Ex 1D

5. Improper Algebraic Fractions

So far we have only considered proper algebraic fractions, i.e. ones where the degree of the numerator is less than the degree of the denominator. If this is not the case, we must decompose the numerator until it is.

Example 1 : Write the following in partial fractions.

Decomposing the numerator,

Note that this is effectively algebraic long division, which we met in units C2 and C3.

1
x2 / + / x / − / 2 / x2 / + / 2x / − / 11
x2 / + / x / − / 2
x / − / 9

We now follow the usual procedure for two linear factors.

Therefore

Example 2 : Write the following in partial fractions.

Now

Again, you can set the working out as algebraic long division if you prefer.

2x / + / 4
x2 / − / 2x / − / 3 / 2x3 / + / 0x2 / − / x / + / 7
2x3 / − / 4x2 / − / 6x
4x2 / + / 5x / + / 7
4x2 / − / 8x / − / 12
13x / + / 19

Comparing numerators,

Therefore

C4 p7 Ex 1E

6. Differentiation Using Partial Fractions

Example 1 : Express in partial fractions. Hence differentiate this function.

We could use the quotient rule from unit C3 to differentiate this as it stands. However, it is much simpler to use partial fractions.

Comparing numerators,

Therefore we have

This expression can now be differentiated using the chain rule, or our quick method which is to remember that .

Example 2 : Express in partial fractions, and hence differentiate it.

Comparing numerators,

Equating coefficients of ,

Therefore

Topic Review : Partial Fractions