PURE MATHEMATICS - 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.

 

The numerators are subsequently 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.

 

 

Example:

equation example

 

 

Identities are satisfied by any value of the variable used.

 

Note the equals sign '=' is modified to reflect this.

 

 

Example:

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

 

 

hence:

 

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

 

 

hence:

repeated factor #3

 

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

 

 

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

hence:

quadratic factors#3

 

 

 

Example #1

quadratic factors#4

 

 

 

 

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