AHL 1.11 (HL)—Partial fractions

Decompose a proper rational expression into partial fractions.

For distinct linear factors, write unknown numerators over each factor, clear denominators, then solve coefficients.

Worked example
(3x+5)/(x(x+2))=A/x+B/(x+2); substituting x=0 gives A=5/2.

Worked example
What condition must hold first? numerator degree is lower than denominator degree.

Common boundary
Partial fractions is not polynomial division unless the fraction is improper.

Complete the decomposition: $\frac{3x+5}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2}$ gives $3x+5=A(x+2)+Bx$. Setting $x=0$ gives $A=5/2$; setting $x=-2$ gives $-1=-2B$, so $B=1/2$. Therefore $\frac{3x+5}{x(x+2)}=\frac{5}{2x}+\frac{1}{2(x+2)}$. Recombine the fractions to check the numerator.