Partial Fraction Decomposition Essentials
Master the key concepts and techniques for breaking down complex rational expressions into simpler fractions.
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Questions Covered in This Set
8 cards to master
What is partial fraction decomposition?
A technique that breaks down complex rational expressions into simpler fractions by reversing the process of adding fractions together.
What is required for a rational expression before applying partial fraction decomposition?
The expression must be proper, meaning the degree of the numerator must be less than the degree of the denominator. If not, perform polynomial long division first.
For distinct linear factors like (x-1)(x+2), what form do the partial fractions take?
Each factor gets its own fraction with a constant numerator: A/(x-1) + B/(x+2).
How do you handle repeated linear factors like (x-1)²?
Include a separate term for each power up to the highest multiplicity: A/(x-1) + B/(x-1)².
What is the strategic substitution method?
A technique where you choose specific x-values (often the zeros of the factors) to eliminate terms and quickly solve for unknown coefficients.
For irreducible quadratic factors like x²+1, what form does the numerator take?
A linear numerator: (Bx+C)/(x²+1), not just a constant.
What is the method of undetermined coefficients?
Expanding all terms and comparing coefficients for each power of x to create a system of equations that solves for the unknown constants.
How can you verify your partial fraction decomposition is correct?
Add your partial fractions back together and simplify; you should get the original expression.
