Decomposition Strategies

Calculus 2 · Axiom Academy

LESSON Decomposition Strategies A strategy for splitting any rational function into simpler fractions — the move that makes hard rational integrals routine. The whole method only works when the fraction is proper — the numerator's degree is strictly below the denominator's. Compare the two top degrees: if the numerator is as big or bigger, long-divide first and decompose only the leftover remainder. Improper: degrees tie, so divide before decomposing 2. Factor the Denominator Completely Once proper, the denominator decides everything. Break Q(x) all the way down into linear factors (ax+b) and irreducible quadratic factors (ax^2+bx+c) that cannot be split further over the real numbers. The list of factors you get is the blueprint for the decomposition. Factor fully — here three distinct linear factors 3. One Term Shape per Factor Type This is the core of the strategy. Each factor contributes its own term, and the shape of the numerator depends on the factor type . Watch the rule build for the three cases, then read them off below. One term for every power 1 to n : . Add up one block from each factor present. A repeated factor needs all powers; a quadratic needs Bx+C With the shape set, multiply both sides by the common denominator and find the unknowns. For distinct linear factors the fastest route is the cover-up method : substitute the x that zeros one factor and every other term vanishes, leaving that constant alone. Watch it isolate A , then B .

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