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

Simplifying:

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.

10x-6/(x+9)(x-3)

This is the answer if we simplify.

Decomposition:

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

10x-6/(x+9)(x-3)

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