Partial Fractions

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Partial fraction decomposition rewrites a proper rational function $\frac{P(x)}{Q(x)}$ (where $\deg P < \deg Q$) as a sum of simpler fractions whose denominators are the linear and irreducible quadratic factors of $Q(x)$.

Four decomposition templates to memorise:

Denominator factor	Numerator template
Distinct linear $(x-a)$	$\dfrac{A}{x-a}$
Repeated linear $(x-a)^k$	$\dfrac{A_1}{x-a} + \dfrac{A_2}{(x-a)^2} + \cdots + \dfrac{A_k}{(x-a)^k}$
Irreducible quadratic $ax^2+bx+c$	$\dfrac{Ax+B}{ax^2+bx+c}$ (always linear numerator)
Repeated irreducible quadratic	$\sum \dfrac{A_i x + B_i}{(ax^2+bx+c)^i}$

Two solving routes: the cover-up (Heaviside) method plugs $x=a$ for distinct linear factors; equating coefficients is used otherwise. ECAT almost always tests this as a setup step for integration of rational functions.

🟡 Standard — Regular Study

Standard content for students with a few days to months.

Setting Up the Decomposition

A rational function $\frac{P(x)}{Q(x)}$ must first be proper — i.e. $\deg P < \deg Q$. If $\deg P \geq \deg Q$, perform polynomial long division first:

$$\frac{P(x)}{Q(x)} = \text{polynomial} + \frac{R(x)}{Q(x)}$$

Only the remainder $R(x)/Q(x)$ can be decomposed. Skipping this step is the most common reason students get a wrong answer on ECAT.

Form of the Decomposition

Let $Q(x)$ factor over the reals as:

$$Q(x) = (x-a_1)(x-a_2)\cdots(x-a_r),(x-b_1)^{m_1}\cdots(ax^2+bx+c)^{n_1}\cdots$$

Then:

$$\frac{P(x)}{Q(x)} = \sum_{i=1}^{r}\frac{A_i}{x-a_i} + \sum_{j=1}^{m_1}\frac{B_j}{(x-b_1)^j} + \sum_{k=1}^{n_1}\frac{C_k x + D_k}{(ax^2+bx+c)^k} + \cdots$$

Solving for the Constants

Cover-up method (Heaviside): For a distinct linear factor $(x-a)$:

$$A = \left.\frac{P(x)}{Q(x)/(x-a)}\right|_{x=a}$$

This instantly gives the constant without expanding any algebra.

Equating coefficients: Multiply through by $Q(x)$, expand, and match coefficients of like powers of $x$. This is required whenever a quadratic or repeated factor appears.

Worked Micro-Example

Decompose $\dfrac{5x+7}{(x+1)(x+3)}$.

Write $\dfrac{5x+7}{(x+1)(x+3)} = \dfrac{A}{x+1} + \dfrac{B}{x+3}$, so $5x+7 = A(x+3)+B(x+1)$.

At $x=-1$: $2 = 2A \Rightarrow A = 1$.
At $x=-3$: $-8 = -2B \Rightarrow B = 4$.

Result: $\dfrac{1}{x+1} + \dfrac{4}{x+3}$.

Why ECAT Tests This

Partial fractions is the gateway technique for integrating functions like $\int \frac{dx}{(x-a)(x-b)}$ and for finding inverse Laplace transforms in engineering mathematics. ECAT items usually present a rational function and ask either for the decomposed form or for the integral built on it. Quadratic factors appear frequently, so remember: the numerator must be linear ($\text{constant}$ alone is wrong).

🔴 Extended — Deep Study

Comprehensive coverage for students on a longer study timeline.

Irreducibility Check

A quadratic $ax^2+bx+c$ is reducible over the reals iff discriminant $D=b^2-4ac \geq 0$. If $D < 0$, it is irreducible and contributes a $\dfrac{Ax+B}{ax^2+bx+c}$ term; if $D \geq 0$, refactor into linear terms and treat those normally. Misclassifying a reducible quadratic wastes a step and usually produces two conflicting answers.

Edge Cases

Repeated irreducible quadratic $(ax^2+bx+c)^2$ requires two terms: $\dfrac{Ax+B}{ax^2+bx+c} + \dfrac{Cx+D}{(ax^2+bx+c)^2}$. Using only one term leaves a degree mismatch.
Degenerate case: If the original numerator and denominator share a common factor, cancel first; otherwise the decomposition has non-existent terms.
Coefficient symmetry: When $Q(x)$ is even or has symmetric roots (e.g. $(x-a)(x+a)$), expect $A$ and $B$ to be equal in magnitude with opposite signs, which can verify a quick answer.

Connection to Integration

After decomposition, the integrals collapse into standard forms:

$\displaystyle\int \frac{dx}{x-a} = \ln|x-a| + C$
$\displaystyle\int \frac{Ax+B}{ax^2+bx+c},dx = \frac{A}{2a}\ln|ax^2+bx+c| + \frac{2Ba-bA}{a\sqrt{4ac-b^2}}\arctan!\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right) + C$

This is why ECAT pairs partial fractions with logarithmic and arctan answers.

Common Mistakes

Decomposing an improper fraction without long division first.
Writing $\dfrac{A}{x^2+1}$ instead of $\dfrac{Ax+B}{x^2+1}$ for the quadratic $x^2+1$.
Dropping the higher-power term on a repeated factor, e.g. omitting $\dfrac{B}{(x-a)^2}$ for $(x-a)^2$.
Sign errors when applying the cover-up method at $x=-a$ versus $x=a$.
Stopping after finding some constants but not verifying by recombining.

Practice Prompts

Decompose $\dfrac{3x^2+2x+1}{(x-1)(x^2+2)}$ and identify each constant using the cover-up for the linear part and equating coefficients for the quadratic part.
Resolve $\dfrac{x+2}{(x+1)^2}$ into partial fractions, then integrate the result explicitly.

Exam Strategy

ECAT allocates roughly 3 % of the MCQ paper to this topic, typically as one direct decomposition question or as a sub-step inside an integration problem. Time budget: under 90 seconds per item when the factorisation is clean; up to 3 minutes when quadratics dominate. Always factor the denominator completely on paper before writing any template — incomplete factorisation is the single largest source of avoidable errors in this topic.

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Partial Fractions