A-Level Maths | Partial Fractions
For distinct linear factors: $\displaystyle\frac{6x}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$
Multiply both sides by $(x-1)(x+2)$, then substitute $x=1$ and $x=-2$ to solve for $A$ and $B$.
Worked Example
$\displaystyle\frac{6x}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$
$6x = A(x+2) + B(x-1)$
$x=1: \;\; 6 = 3A \;\Rightarrow\; A = 2$
$x=-2: \;\; -12 = -3B \;\Rightarrow\; B = 4$
$\displaystyle\frac{6x}{(x-1)(x+2)} = \frac{2}{x-1} + \frac{4}{x+2}$
Question 1
Express each of the following as partial fractions.
a) $\displaystyle\frac{6x}{(x-1)(x+2)}$
b) $\displaystyle\frac{7y-11}{(y+2)(y-3)}$
c) $\displaystyle\frac{19-4t}{(t+4)(t-3)}$
d) $\displaystyle\frac{w-22}{(w+2)(w-6)}$
e) $\displaystyle\frac{8z+7}{(z+2)(z-7)}$
Show Answers
a) $\displaystyle\frac{2}{x-1} + \frac{4}{x+2}$
b) $\displaystyle\frac{2}{y-3} + \frac{5}{y+2}$
c) $\displaystyle\frac{1}{t-3} - \frac{5}{t+4}$
d) $\displaystyle\frac{3}{w+2} - \frac{2}{w-6}$
e) $\displaystyle\frac{1}{z+2} + \frac{7}{z-7}$
For a repeated linear factor: $\displaystyle\frac{6x}{(x-2)(x+1)^2} = \frac{A}{x-2} + \frac{B}{x+1} + \frac{C}{(x+1)^2}$
Multiply both sides by $(x-2)(x+1)^2$, substitute convenient values of $x$, then compare coefficients or substitute $x=0$ for any remaining unknown.
Worked Example
$\displaystyle\frac{4x+1}{(x-2)(x+1)^2} = \frac{A}{x-2} + \frac{B}{x+1} + \frac{C}{(x+1)^2}$
$4x+1 = A(x+1)^2 + B(x-2)(x+1) + C(x-2)$
$x=2: \;\; 9 = 9A \;\Rightarrow\; A = 1$
$x=-1: \;\; -3 = -3C \;\Rightarrow\; C = 1$
$x=0: \;\; 1 = A - 2B - 2C \;\Rightarrow\; 1 = 1 - 2B - 2 \;\Rightarrow\; B = -1$
$\displaystyle\frac{4x+1}{(x-2)(x+1)^2} = \frac{1}{x-2} - \frac{1}{x+1} + \frac{1}{(x+1)^2}$
Question 2
Express each of the following as partial fractions.
a) $\displaystyle\frac{2x^2-x-3}{(x-2)(x-1)^2}$
b) $\displaystyle\frac{y^2-2y+8}{(y+2)(y-2)^2}$
c) $\displaystyle\frac{-3t^2+12t+7}{(t-3)(t+1)^2}$
d) $\displaystyle\frac{-3w^2+10w-11}{(w-2)(w-1)^2}$
Show Answers
a) $\displaystyle\frac{3}{x-2} - \frac{1}{x-1} + \frac{2}{(x-1)^2}$
b) $\displaystyle\frac{1}{y+2} + \frac{2}{(y-2)^2}$
c) $\displaystyle\frac{1}{t-3} - \frac{4}{t+1} + \frac{2}{(t+1)^2}$
d) $\displaystyle\frac{4}{(w-1)^2} - \frac{3}{w-2}$
Worked Example
$\displaystyle\frac{2y+3}{y^2(y+1)} = \frac{A}{y} + \frac{B}{y^2} + \frac{C}{y+1}$
$2y+3 = Ay(y+1) + B(y+1) + Cy^2$
$y=0: \;\; 3 = B \;\Rightarrow\; B = 3$
$y=-1: \;\; 1 = C \;\Rightarrow\; C = 1$
Compare $y^2$: $0 = A + C \;\Rightarrow\; A = -1$
$\displaystyle\frac{2y+3}{y^2(y+1)} = -\frac{1}{y} + \frac{3}{y^2} + \frac{1}{y+1}$
Question 3
Express each of the following as partial fractions.
a) $\displaystyle\frac{2x^2-3}{(3-2x)(1-x)^2}$
b) $\displaystyle\frac{3y^2+17y+4}{y^2(y+4)}$
c) $\displaystyle\frac{t^2}{(t-2)(t-1)^2}$
d) $\displaystyle\frac{9w^2}{(2w+1)(w-1)^2}$
Show Answers
a) $\displaystyle\frac{6}{3-2x} - \frac{2}{1-x} - \frac{1}{(1-x)^2}$
b) $\displaystyle\frac{1}{y^2} + \frac{4}{y} - \frac{1}{y+4}$
c) $\displaystyle\frac{4}{t-2} - \frac{1}{(t-1)^2} - \frac{3}{t-1}$
d) $\displaystyle\frac{4}{w-1} + \frac{3}{(w-1)^2} + \frac{1}{2w+1}$
If the degree of the numerator is $\geq$ the degree of the denominator, first divide the numerator by the denominator: $\displaystyle\frac{f(x)}{g(x)} = Q(x) + \frac{R(x)}{g(x)}$, where $Q(x)$ is the quotient and $R(x)$ the remainder.
Worked Example
$\displaystyle\frac{x^3+3x^2-5}{x+2}$
$x^3+3x^2-5 = (x+2)(x^2+x-2) - 1$
$\displaystyle\frac{x^3+3x^2-5}{x+2} = x^2+x-2 - \frac{1}{x+2}$
Question 4
For each of the following, find the constants $a$, $b$, $c$ and $d$ (or $a$, $b$, $c$ where only three appear).
a) $\displaystyle\frac{x^3+2x^2+3x-4}{x+1} \equiv ax^2+bx+c+\frac{d}{x+1}$
b) $\displaystyle\frac{2x^3+3x^2-4x+5}{x+3} \equiv ax^2+bx+c+\frac{d}{x+3}$
c) $\displaystyle\frac{2x^2+4x+5}{x^2-1} \equiv a + \frac{bx+c}{x^2-1}$
Show Answers
a) $a=1,\; b=1,\; c=2,\; d=-6$
b) $a=2,\; b=-3,\; c=5,\; d=-10$
c) $a=2,\; b=4,\; c=7$
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