Lesson4 Partial Fractions

8/3/2019 Lesson4 Partial Fractions

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Partial Fractions

Learning Objectives Use partial fraction terminology

Identify types of partial fractions

Express complex looking algebraic

equations in terms of two manageable terms


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Terminology

The following definitions will help make this section clearer.

Polynomial of degree n.

If P(x) = anxn + an-1x

n-1 + an-2xn-2 ++ a2x

2 + a1x1 + a0

Where an,., a0 RThen P is a polynomial of degree n

Examples

x + 3 has degree1, x2

+4x+1 has degree 2, 3x4

+ 6x2

+

x+5 hasdegree 4. Constants such as 5 have degree 0.


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Rational function

If P(x) and Q(x) are polynomials then is called a

rational function.

Proper rational function

Let P(x) be a polynomial of degree n and Q(x) be a polynomial

of degree m.

If n < m then is a proper rational function.

)(

)(

xQ

xP

)(

)(

xQ

xP

Improper rational functionLet P(x) be a polynomial of degree n and Q(x) be a polynomial

of degree m.

If n m then is an improper rational function.)(

)(

xQ

xP


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Types of Partial Fractions

The process of taking a proper rational function and splitting it

into separate terms each with a factor of the original denominator

as its denominator is called expressing the function in

partial fractions.

Putting fractions over a common denominator, for example,

2 11 1 2 3

1 2 1 2 1 2

x x x

x x x x x x

Is a familiar process.

The opposite process of expressing

2 3 1 1

as1 2 1 2

x

x x x x

Is called putting a proper rational

Function into partial fractions.


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The way in which the rational function splits up depends on whether

the denominator is a quadratic equation or a cubic equation and

Whether it has linear, repeated linear or quadratic factors (with no

real roots).

Type 1 Linear or constant

quadratic

Type 1a

... A B

x a x b x a x b

This has denominator of a quadratic with two distinct roots

ExampleExpress as partial fractions 2

5

5 6

x

x x


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Solution

2

5 5

2 35 6

x x

x xx x

5

Let2 3 2 3

x A B

x x x x

3 2

2 3

A x B x

x x

5 3 2x A x B x

1 A B

5 3 2A B

3 2A B

2

5 3 2

2 35 6

x

x xx x

1) Factorise the denominator

2) Identify the type of partial fraction

3) Obtain the fractions with a

common denominator

4) Equate numerators since the

denominators are equal.

5) Equate coefficients of powers of x

6) Solve equations simultaneously


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Type 1b

2 2

.... A B

x ax a x a

This has a denominator of a quadratic with repeated factors.

ExampleExpress in partial fractions.

2

11 3

3

x

x

Solution

2 2

11 3Let33 3

x A Bxx x

2

3

3

A x B

x

11 3 3x A x B 3

11 3

A

B A

3, 2A B 2 2

11 3 3 2

33 3

x A B

xx x

Step 2 notice step 1 not needed

Step 3

Step 4

Step 5: equating powers of x and constants

Step 6

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Lesson4 Partial Fractions