Partial fractions

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1. 1Partial fractions with linear factorsSplit a fraction whose denominator has two or three different linear factors, find where it is undefined, and divide first when the top is not of lower degree.Done18 min
2. 2Repeated and quadratic factorsPartial fractions when a bracket is repeated, such as (x + 2)³, or when a quadratic factor won't factorise, such as x² + 2: which fractions to write, and how to find the constants.Done18 min

Past questions (20)

From 2008–2023: 17 WAEC, 3 NECO. Theory questions first, then objective, newest first.

Theory (16)

• NECO 2023 · Paper 2 · Q10Resolve $\dfrac{2x^3 + 5x^2 - 6x + 4}{(x - 1)(x + 2)}$ into partial fractions.
• NECO 2023 · Paper 2 · Q11Solve the equation $\dfrac{1}{x + 1} + \dfrac{2}{x - 1} = \dfrac{1}{x + 3}$ (2 d.p.).
• WAEC 2023 · Paper 2 · Q9Two functions $g$ and $h$ are defined on the set of real numbers, $\mathbb R$, by $g: x \to \dfrac{2 + x}{x^2 + x - 20}$ and $h: x \to x + 3$.
• WAEC 2022 · Paper 2 · Q10Resolve $\dfrac{x^3 - x^2 - 4}{x^2 - 1}$ into partial fractions.
• WAEC 2019 · Paper 2 · Q10The lengths of the sides of a triangle, in centimetres, are $(3 + \sqrt2)$, $2\sqrt3$ and $3\sqrt2$. Find the value of the largest angle of the …
• WAEC 2018 · Paper 2 · Q3Using the substitution $u = x - 2$, write $\dfrac{x^3 + 5}{(x - 2)^4}$ as an expression in terms of $u$.
• WAEC 2018 · Paper 2 · Q10The function $f(x) = px^2 + qx + r$, where $p$, $q$ and $r$ are constants. If $f(1) = 0$, $f(-1) = 4$ and $f(2) = 7$, find the: (i) values of …
• WAEC 2017 · Paper 2 · Q10If $f(x) = \dfrac{2x - 3}{(x^2 - 1)(x + 2)}$, (i) find the values of $x$ for which $f(x)$ is undefined; (ii) express $f(x)$ in partial fractions.
• WAEC 2017 · Paper 2 · Q2Evaluate $\displaystyle\int_1^3 \frac{x - 1}{(x + 1)^2}\,dx$ (3 d.p.).
• WAEC 2016 · Paper 2 · Q9Express $\dfrac{x + 6}{(x + 1)^3}$ in partial fractions.
• WAEC 2014 · Paper 2 · Q9Resolve $\dfrac{x^2 + 1}{(x + 2)^3}$ into partial fractions.
• WAEC 2011 · Paper 2 · Q12Find the equation of the tangent to the curve $\dfrac{x^2}{4} + y^2 = 1$ at the point $\left(1, \dfrac{\sqrt3}{2}\right)$.
• WAEC 2011 · Paper 2 · Q11Solve for $x$, $y$ and $z$ in the equations: $3x + 5y - 4z = -5$; $6x + 3y - 5z = 26$; $-2x + 2y + z = -11$.
• WAEC 2009 · Paper 2 · Q9Express $\dfrac{4x^2 - x + 3}{(x^2 + 1)(x - 1)}$ in partial fractions.
• WAEC 2008 · Paper 2 · Q11Express $\dfrac{2x^2 - 5x + 1}{x^3 - 4x^2 + 3x}$ in partial fractions.
• WAEC 2008 · Paper 2 · Q11Find $\dfrac{dy}{dx}$ if $x^2 + 4xy - y^2 = 7$.

Objective (4)

• NECO 2023 · Paper 1 · Q12Resolve $\dfrac{2x + 5}{(x + 3)(x + 2)}$ into partial fractions.
• WAEC 2023 · Paper 1 · Q7Given that $\dfrac{3x + 4}{(x - 2)(x + 3)} \equiv \dfrac{P}{x + 3} + \dfrac{Q}{x - 2}$, find the value of $Q$.
• WAEC 2022 · Paper 1 · Q8Given that $\dfrac{8x + m}{x^2 - 3x - 4} \equiv \dfrac{5}{x + 1} + \dfrac{3}{x - 4}$, find the value of $m$.
• WAEC 2020 · Paper 1 · Q8If $\dfrac{6x + k}{2x^2 + 7x - 15} \equiv \dfrac{4}{x + 5} - \dfrac{2}{2x - 3}$, find the value of $k$.