Partial Fractions

Introduced to in Math 117

Rules

If Q factors into a product of n distinct linear factors, Q(x) = (x-a $_{1}$ )(x-a $_{2}$ ) … (x-a $_{n}$ ) , then $\frac{P ( x )}{Q ( x )}$ = $\frac{A _{1}}{x - a _{1}}$ + $\frac{A _{2}}{x - a _{2}}$ + … $\frac{A _{n}}{x - a _{n}}$
If Q has a quadratic factor (x $^{2}$ + bx + c), then decomposition contains the term: $\frac{BX + C}{x ^{2} + b x + c}$
If (x-a) $^{m}$ is a repeated linear factor of Q then the decomposition contains the terms: $\frac{A _{1}}{x - a}$ + $\frac{A _{2}}{( x - a ) ^{2}}$ + … + $\frac{A _{m}}{( x - a ) ^{m}}$
If (x $^{2}$ + bx + c) $^{k}$ is a repeated quadratic factor of Q, then the decomposition contains the terms: $\frac{B _{1} X + C _{1}}{x ^{2} + b x + c}$ + $\frac{B _{2} X + C _{2}}{( x ^{2} + b x + c ) ^{2}}$ … $\frac{B _{k} X + C _{k}}{( x ^{2} + b x + c ) ^{k}}$
It is always possible to factor Q(x) into a product linear factors and/or quadratic factors