is a powerful tool for simplifying complex rational expressions. It breaks down tricky fractions into simpler ones, making them easier to work with in calculus and beyond.
This technique is super useful for integration and solving differential equations. By mastering partial fractions, you'll be able to tackle more advanced math problems with confidence and ease.
Partial Fraction Decomposition
Decomposition of rational expressions
Breaks down complex into sum of simpler fractions
For nonrepeated linear factors in denominator, takes form:
(x−a)(x−b)(x−c)P(x)=x−aA+x−bB+x−cC
Find values of A, B, C by multiplying both sides by common denominator and equating coefficients or evaluating at specific points
Example: (x−1)(x+2)2x+1=x−1A+x+2B
Multiply both sides by (x−1)(x+2): 2x+1=A(x+2)+B(x−1)
Equate coefficients or evaluate at x=1 and x=−2 to solve for A and B
Useful for simplifying complex fractions into more manageable terms (integration, differential equations)
Partial fractions with repeated factors
For repeated linear factors in denominator, decomposition includes terms with powers of repeated factor