Partial Fraction Decomposition

Decomposition Setup

(3x + 5) / ((x + 1)(x + 2))

Form: P(x) / ((ax + b)(cx + d)) = A/(ax + b) + B/(cx + d)

Numerator coefficients (degree ≤ 1)

Coeff of x

Constant

Factor 1: (ax + b)

Factor 2: (cx + d)

Examples

(x + 3) / (x + 1)²

Form: P(x) / (ax + b)² = A/(ax + b) + B/(ax + b)²

Numerator coefficients (degree ≤ 1)

Coeff of x

Constant

Repeated factor: (ax + b)²

Examples

(2x² + x + 3) / ((x + 1)(x² + 1))

Form: P(x) / ((x + r)(x² + q)) = A/(x + r) + (Bx + C)/(x² + q)

Numerator coefficients (degree ≤ 2)

x²

const

Linear factor: (x + r)

Irreducible quadratic: (x² + q), q > 0

Examples

Decomposition

Enter a rational expression and press Decompose to see the partial fraction form, coefficients A, B, C, and a verification check.

Decomposed Form

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Coefficients

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Verification (recombine)

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When to Use Each Form

Distinct linear: A/(ax + b) Repeated linear: A/(ax+b) + B/(ax+b)² Irred. quadratic: (Ax + B)/(x² + q)

Distinct linear factors: Each unique factor (ax + b) in Q(x) contributes one term A/(ax + b) to the decomposition.

Repeated linear factors: If (ax + b) appears twice, write A/(ax + b) + B/(ax + b)². For n repetitions, write terms up to power n.

Irreducible quadratic factors: If x² + q cannot be factored over the reals (q > 0), its numerator must be linear: (Bx + C)/(x² + q).

Always check deg(P) < deg(Q) first. If not, perform polynomial long division before decomposing.

Cover-Up Method (Distinct Linear Only)

A = P(x) / Q(x) |_{x = root of factor}

For distinct linear factors only, you can find each coefficient instantly by covering the corresponding factor in the denominator and substituting the factor's root into the rest of the expression.

Example: For (3x+5)/((x+1)(x+2)):

A: cover (x+1), set x = −1 → (3(−1)+5)/(−1+2) = 2/1 = 2

B: cover (x+2), set x = −2 → (3(−2)+5)/(−2+1) = (−1)/(−1) = 1

This shortcut only works for distinct linear factors. Repeated or quadratic factors require coefficient matching.

Root of (ax + b) is x = −b/a.
Cover-up gives A directly — no system of equations needed.
Verify by recombining the partial fractions.
For quadratic factors, always match x² and x coefficients separately.

Partial Fraction Decomposition

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