copyright © 2011 pearson education, inc. partial fractions section 5.4 systems of equations and...
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Copyright © 2011 Pearson Education, Inc.
Partial FractionsSection 5.4
Systems of Equations and Inequalities
Copyright © 2011 Pearson Education, Inc. Slide 5-3
5.4
We have learned how to add rational expressions. Now we will reverse the process of addition.
We start with a rational expression and write it as a sum of two or more simpler rational expressions, called partial fractions.
An equation consisting of the sum of partial fractions on the right-hand-side is called the partial fraction decomposition of the rational expression on the left-hand-side.
Partial Fraction Decomposition
Copyright © 2011 Pearson Education, Inc. Slide 5-4
5.4
In general, let N(x) be the polynomial in the numerator and D(x) be the polynomial in the denominator of the fraction that is to be decomposed. We will decompose only fractions for which the degree of the
numerator is smaller than the degree of the denominator. If the degree of N(x) is not smaller than the degree of D(x), we can
use long division to write the rational expression as quotient + remainder/divisor. If a factor of D(x) is repeated n times, then all powers of the factor
from 1 through n might occur as denominators in the partial fractions.
To find the partial fraction decomposition of a rational expression, the denominator must be factored into a product of prime polynomials. If a quadratic prime polynomial occurs in the denominator, then
the numerator of the partial fraction for that polynomial is of the form Ax + B.
General Decomposition
Copyright © 2011 Pearson Education, Inc. Slide 5-5
5.4
1. If the degree of the numerator N(x) is greater than or equal to the degree of the denominator D(x), use division to express N(x)/D(x) as quotient + remainder/divisor and decompose the resulting fraction.
2. If the degree of N(x) is less than the degree of D(x), factor the denominator completely into prime factors that are either linear (ax+b) or quadratic (ax2+bx+c).
3. For each linear factor of the form (ax+b)n, the partial fraction decomposition must include the following fractions:
4. For each quadratic factor of the form (ax2+bx+c)m, the partial fraction decomposition must include the following fractions:
5. Set up and solve a system of equations involving the As, Bs, and/or Cs.
nn
bax
A
bax
A
bax
A
)(...
)( 221
mmm
cbxax
CxB
cbxax
CxB
cbxax
CxB
)(...
)( 22222
211
Strategy: Decompositioninto Partial Fractions