Thursday, 24 January 2013

Partial fractions summary (Shaun Ng)

Intro:  Partial fractions

- Instead of simplifying two fractions into one, we want to break them up here.
- Degree is the highest power of x in the set
- Proper fractions = Numerator < Denominator
- Improper fractions = Numerator > Denominator
- Proper algebraic fractions = Degree of numerator < Degree of Denominator
- Improper algebraic fractions = Degree of numerator > Degree of Denominator


Previously we have been taught how to simplify two fractions into one. (/ sign = border between numerator and denominator)
For example:

8/x+9 + 2/x-3

1st step: Change denominator to the same thing, to do this multiple both numerator and denominator by the denominator in the fraction it is added or subtracted to.

8x-24/(x+9)(x-3) + 2x+18/(x+9)(x-3)

2nd step: Add or subtract the fractions together, by adding the numerators into one fraction.


This is the answer if we simplify.


Now lets try something new using the same equation from example of simplifying.


1st step: Separate into two different fractions with a variable as numerator.

A1/x+9 + A2/x-3

2nd step: Multiply through the bottom to eliminate denominator

A1(x-3) + A2(x+9)

3rd step: Bring the 10x+6 in

10x+6 = A1(x-3) + A2(x+9)

4th step: Find the roots, so that we can eliminate 1 variable and find the other

Root for x-3 is 3
Root for x+9 is -9

5th step: Sub in and solve

Let x be 3
10(3) + 6 = A1(3-3) + A2(3+9)
36 = A2(12)
A2 = 3

Let x be -9
10(-9) + 6 = A1(-9-3) + 3(-9+9)
-96 = A1(-12)
A1 = 8

Hence, A1 = 8 and A2 = 3

6th step: Express as partial fraction

8/x+9 + 3/x-3

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