Partial Fractions.notebook

October 25, 2013

Partial Fractions

1

Partial Fractions.notebook

October 25, 2013

Try adding these fractions using a common denominator:

In reverse, this is known as rewriting a fraction using PARTIAL FRACTIONS:

How can we find the numerators and denominators of the partial fractions? Partial fractions can be used to aid differentiation and integration.

2

Partial Fractions.notebook

October 25, 2013

DISTINCT LINEAR FACTORS If the denominator contains DISTINCT LINEAR FACTORS (as before) then the partial fractions are of the form

for some constants A and B.

We can use algebra to work out the constants A and B.

3

Partial Fractions.notebook

October 25, 2013

Example:Express in partial fractions.

4

Partial Fractions.notebook

October 25, 2013

Example:Express in partial fractions.

TEXTBOOK P18 EX2 5

Partial Fractions.notebook

October 25, 2013

REPEATED LINEAR FACTORS Try adding these fractions using a common denominator:

In reverse we see that if the fraction has a repeated linear factor then we need to consider and include two DIFFERENT partial fractions for the factor. If the denominator contains REPEATED LINEAR FACTORS then the partial fractions are of the form

for some constants A and B. 6

Partial Fractions.notebook

October 25, 2013

Example:Express in partial fractions.

TEXTBOOK P19 EX3

7

Partial Fractions.notebook

October 25, 2013

IRREDUCIBLE QUADRATIC FACTOR A quadratic factor is irreducible if it cannot be factorised (use the discriminant as proof!).

If the denominator contains an IRREDUCIBLE QUADRATIC FACTOR then the corresponding partial fraction is of the form:

for some constants A and B.

8

Partial Fractions.notebook

October 25, 2013

Example:Express in partial fractions.

9

Partial Fractions.notebook

October 25, 2013

Example:Express in partial fractions.

TEXTBOOK P20 EX4

10

Partial Fractions.notebook

October 25, 2013

ALGEBRAIC LONG DIVISION An algebraic improper rational function is one where the degree of the numerator is greater than or equal to the degree of the denominator. Eg:

Simplifying the last fraction shows that an improper rational function can be expressed as the sum of a polynomial and a proper rational function:

If the denominator is more complex we must use algebraic long division to simplify the fraction.

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Partial Fractions.notebook

October 25, 2013

Example:Simplify:

12

Partial Fractions.notebook

October 25, 2013

Example:Simplify:

13

Partial Fractions.notebook

October 25, 2013

Example:Simplify:

14

Partial Fractions.notebook

October 25, 2013

Example:Simplify:

TEXTBOOK P17 EX1

15

Partial Fractions.notebook

October 25, 2013

Example:Express as the sum of a polynomial and partial fractions:

TEXTBOOK P22 EX6 16

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