Algebra: Partial Fractions

[ definitions ][ identities ][ linear ]

[ repeated factors][ quadratic factor]




some definitions:

Proper Fraction When the degree(index) of the function is higher in the denominator than the numerator.

Improper Fraction When the degree(index) of the function is higher in the numerator than the denominator.

Partial Fractions Factorising the denominator of a proper fraction means that the fraction can be expressed as the sum(or difference) of other proper fractions.


Simple addition/subtraction of algebraic fractions

As with simple fraction arithmetic, a common denominator is found from the denominators of either fraction and the numerators altered to be fractions of the new denominator.

simple algebraic fraction addition

algebraic fraction addition #2

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Equations & Identities

Equations are satisfied by discrete values of the variable involved.


equation example

Identities are satisfied by any value of the variable used. Note the equals sign '=' is modified to reflect this.


identity example

When we make partial fractions(below) we are creating an identity from the original expression.

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Denominator with only 'linear factors'

By 'linear' we mean that x has a power no higher than '1' . In other words, this method does not work with x2, x3, x4 etc.

For each linear factor of the type:

linear type #1

there is a partial fraction:

linear #2


linear #3

where x is a variable and A,B,a,b,c,d are constants, where 'a' is not equal to 'b'.

Example #1

partial fractions example#1

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Denominator with 'repeated' linear factors

For each 'repeated' linear factor of the type:

repeated linear factors #1

there is a partial fraction:

repeated linear factor #2


repeated factor #3

Example #1

repeated factors#4

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Denominator with a quadratic factor

For each quadratic factor of the type:

quadratic factors#1

there is a partial fraction:

quadratic factors #2


quadratic factors#3

Example #1

quadratic factors#4



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