math academy-partial-fractions-notes

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Having a positive mental attitude is asking how something can be done rather than saying it can't be done.

Bo Bennett quotes

Notes: Partial Fractions Introduction We have learnt how to combine fractions as follows:

)2)(1(12

22

11

−+

−=

−+

+ xxx

xx

Partial Fractions is the reverse process of splitting a single fraction into a sum of two or more fractions, i.e

22

11

)2)(1(12

−+

+=

−+

−

xxxxx

[A] Polynomial

A polynomial in one variable x is given by

012

21

1 ... axaxaxaxa nn

nn +++++ −

−

where 0121 ,,,, aaaaa nn − are constants and n is a non negative integer.

If 0≠na , then the polynomial has degree n .

Egs (i) 5432 24 +++ xxx is a polynomial of degree 4

(ii) 23 16x

x − is not a polynomial

[B] Rational Function

A rational function is an expression of the form )()(xQxP

where )(xP and )(xQ are

polynomials.

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[C] Proper and Improper Function

The rational function )()(xQxP

is a

(i) Proper Fraction if the degree of )(xP < degree of )(xQ

4

3 53xxx −+

(ii) Improper Fraction if the degree of )(xP ≥ degree of )(xQ

2

2 53xxx −+ ,

125

34

6

+− xxx

Case 1: Linear Factor (ax+b) To every linear factor (ax+b) in the denominator of a proper fraction, there corresponds

a partial fraction of the form bax

A+

.

Example 1: Express )3)(2(

12++

+

xxx

in partial fractions.

[3 methods: Substitution, Comparing coefficients and Cover up Rule] ws 1 Q1

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Case 2 : Repeated Linear Factor (ax+b) To every repeated linear factor (ax+b) repeated n times in the denominator of a proper fraction, there corresponds a sum of n partial fractions:

nn

baxA

baxA

baxA

)(...

)( 221

+++

++

+

Example 2: Express 2)32(12

−

−

xx

in partial fractions

ws 1 Q2

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Case 3: Quadratic Factor (ax 2 +bx+c) which cannot be factorised To every quadratic factor (ax 2 +bx+c) in the denominator of a proper fraction, there

corresponds a partial fraction of the form cbxax

BAx++

+2 .

Example 3: Express )2)(12(

152

2

++

+

xxx in partial fractions

ws 1 Q3

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Two Important Checks Before Starting on Any Question Check 1: (Must be a Proper Fraction) Otherwise Apply Long Division First

Example 4: Express 542

3

−− xxx in partial fractions

Check 2: (Denominator Must Be Completely Factorised)

Example 5: Express 65

122 ++

+

xxx

in partial fractions

ws 1 Q4,5 END