8.10. Partial Fractions http://www.ck12.org
8.10 Partial Fractions
Here you will apply what you know about systems and matrices to decompose rational expressions into the sum of
several partial fractions.
When given a rational expression likex^42 x−− 39 xit is very helpful in calculus to be able to write it as the sum of two
simpler fractions like^3 x+x−^13. The challenging part is trying to get from the initial rational expression to the
simpler fractions.
You may know how to add fractions and go from two or more separate fractions to a single fraction, but how do you
go the other way around?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/61697http://www.youtube.com/watch?v=S-XKGBesRzk Khan Academy: Partial Fraction Expansion 1
Guidance
Partial fraction decompositionis a procedure that reverses adding fractions with unlike denominators. The most
challenging part is coming up with the denominators of each individual partial fraction. See if you can spot the
pattern.
x^2 (x−^61 x−)(^1 x^2 + 2 )=Ax+xB^2 +xC−^1 +Dxx^2 ++E 2
In this example each individual factor must be represented. Linear factors that are raised to a power greater than
one must have each successive power included as a separate denominator. Quadratic terms that do not factor to be
linear terms are included with a numerator that is a linear function ofx.
Example A
Use partial fractions to decompose the following rational expression.
7 xx^23 ++x 3 +x 6
Solution: First factor the denominator and identify the denominators of the partial fractions.
(^7) xx(^2 x+ (^2) +x+ 3 ) (^6) =Ax+Bxx 2 ++C 3
When the fractions are eliminated by multiplying through by the LCD the equation becomes:
7 x^2 +x+ 6 =A(x^2 + 3 )+x(Bx+C)
7 x^2 +x+ 6 =Ax^2 + 3 A+Bx^2 +Cx