Engng Math 145 Partial Fractions

Degree of numerator < denominator

No long division required and partial fractions can be applied immediately.

Polynomial long division

If degree of numerator greater than degree of denominator, then do long division.

• Divide dividend by the x term if divisor and write quotient

• Multiply quotient by the same x term and subtract it from dividend

• Add dividend of term one lower than the quotient’s current degree

• Repeat process but on the “difference + dividend term” until degree of the difference term is lower than the degree of the divisor

• Remainder is remainder divided by the divisor

Polynomial long division shortcut for equal degrees

Divide leading terms for quotient (y) and then subtract dividend by divisor times quotient (y) for remainder.

eg. (2x² + 2x + 1) / (x² + x + 1) → 2x² / x² = 2, (2x² + 2x + 1) - 2(x² + x + 1) = -1 → 2 + (-1) / (x² + x + 1) [answer]

Distinct linear factors

Repeated linear factors

A₁, A₂, …, are unique constants A, B, C, D, …

Distinct irreducible quadratic factors

Repeated quadratic factors

A₁, A₂, … & B₁, B₂, … are unique constants A, B, C, D, …

Integral of 1 / (x² + a²)

Irreducible quadratic test

If discriminant < 0, then irreducible.

Solving partial fraction constants

Simultaneous equations OR equating coefficients.

If simulatneous not possible make a variable equal to i = sqrt(-1).

xⁿ factor

xⁿ is a repeating linear factor: (x - 0)ⁿ