Partial Fraction Decomposition?

Question

What exactly is a partial fraction anyway?

What does the term "Partial Fraction Decomposition" mean anyway?

Write the partial fraction decomposition of each rational expression.

(1) 3x/(x + 2)(x - 1)

(2) (x^5 + 1)/(x^6 - x^4)

Answer 1

The technique of partial fraction means that the rational function is expressed as a sum of parts of the factors of the denominator. This technique is unique and solvable whenever the numerator has degree less than the denominator. Otherwise, apply the division algorithm first.



There are set formulas for these functions. The denominator must be factored completely first.



1. 3x/(x+2)(x-1) = A/(x+2) + B/(x-1)

cross multiply...

3x = A(x-1) + B(x+2)



if x = 1: 3 = 3B ... B = 1



if x = -2: -6 = -3A .. A = 2



Thus 3x/(x+2)(x-1) = 2/(x+2) + 1/(x-1)



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2. (x^5+1) / x^4(x-1)(x+1) = A/x^4 + B/x^3 + C/x^2 + D/x + E/(x-1) + F/(x+1)

cross multiply...



x^5 + 1 = A(x-1)(x+1) + Bx(x+1)(x-1) + Cx^2(x-1)(x+1) + Dx^3(x-1)(x+1) + Ex^4(x+1) + Fx^4(x-1)



we can simply compare coefficients...



x^5 + 1 = A(x^2-1) + B(x^3-x) + C(x^4-x^2) + D(x^5-x^3) + E(x^5+x^4) + F(x^5-x^4)



= [D+E+F]x^5 + [C+E-F]x^4 + [B-D]x^3 + [A-C]x^2 + [-B]x - A



thus...

A = -1 .. B = 0 .. C = -1 .. D = 0 ..

E+F = 1 & E-F=0 ...... E = 1/2 ... F = 1/2



(x^5 + 1)/(x^6 - x^4) = -1/x^4 - 1/x^2 + 1/{2(x-1)} + 1/{2(x+1)}

Answer 2

The above answers are correct. Partial fraction decomposition is also important in integration. Since integration cannot be done directly to rational expressions (polynomial divided by a polynomial), you have to split the rational expression into partial fractions and then integrate each piece. If you go on to calculus Partial Fractions will come up again.

Answer 3

When a fraction is presented to you, look at the denominator.

most of the time it is made up of one or more multiplied factors. Well that was a result of adding two or more original fractions and you need to find the original added fractions.



there are tow ways to do that. The long way and the "cover up" method.



the long way is:

Breakup the fraction supposing that one numerator is A and the other is B like this:



A/(x+2) + B/(x-1)



then find the common denominator of the added fractions but only process the numerator. Then set it equal to the given numerator and solve the resulting equations for the unknown numerators.



Ax - A + Bx +2B = x(A+B) = 3x

A+B=3

2B-A=0 2B=A

so A = 2 and B=1



2/(x+2) + 1/(x-1)

Now find the common denominator of these fractions and combine them to get the one you started with to check your answer.