Algebra

CIE A-Level Mathematics· Pure 3: 1 Algebra· 5 min read

1. Proper and Improper Rational Functions★★☆☆☆⏱ 10 min

📘 Definition

Rational Function

f(x) = \frac{P(x)}{Q(x)}

A ratio of two polynomials $P (x)$ (numerator) and non-zero $Q (x)$ (denominator). Classified by the degrees of the two polynomials.

Example:

$\frac{3 x ^{2} + 2 x + 1}{x ^{2} - 4}$ is improper; $\frac{2 x + 5}{x ^{2} - 4}$ is proper

Any improper rational function must first be simplified to a polynomial plus a proper rational function before partial fractions or expansion can be applied. This is done via polynomial long division or algebraic equating of coefficients.

📐 Worked Example

Simplify $\frac{3 x ^{2} + 2 x - 2}{x ^{2} + x - 2}$ into a polynomial plus a proper rational function.

1
Compare leading terms: $\frac{3 x ^{2}}{x ^{2}} = 3$ . Multiply the denominator by 3:
2
$3(x^2 + x - 2) = 3x^2 + 3x - 6$
3
Subtract this product from the original numerator:
4
$(3x^2 + 2x - 2) - (3x^2 + 3x - 6) = -x + 4$
5
Rewrite the original function:
6
$\frac{3x^2 + 2x - 2}{x^2 + x - 2} = 3 + \frac{-x + 4}{x^2 + x - 2}$

2. Partial Fraction Decomposition★★★☆☆⏱ 20 min

Partial fraction decomposition breaks a proper rational function into a sum of simpler fractions, which is required for integration and series expansion. The form of the decomposition depends entirely on the factors of the denominator:

Distinct linear factor $(a x + b)$ : term $\frac{A}{a x + b}$
Repeated linear factor $(a x + b)^{2}$ : terms $\frac{A}{a x + b} + \frac{B}{( a x + b ) ^{2}}$
Irreducible quadratic factor $(a x^{2} + b x + c)$ : term $\frac{A x + B}{a x ^{2} + b x + c}$

📐 Worked Example

Decompose $\frac{- x + 4}{x ^{2} + x - 2}$ into partial fractions.

1
Factorise the denominator:
2
$x^2 + x - 2 = (x + 2)(x - 1)$
3
Set up the partial fraction form for two distinct linear factors:
4
$\frac{-x + 4}{(x + 2)(x - 1)} = \frac{A}{x + 2} + \frac{B}{x - 1}$
5
Multiply both sides by $(x + 2) (x - 1)$ to eliminate denominators:
6
$-x + 4 = A(x - 1) + B(x + 2)$
7
Substitute the roots of the denominator to find constants: $x = 1$ gives $3 = 3 B ⟹ B = 1$ ; $x = - 2$ gives $6 = - 3 A ⟹ A = - 2$
8
Write the final decomposition:
9
$\frac{-2}{x + 2} + \frac{1}{x - 1}$

Exam tip:

Always verify your answer by combining partial fractions back into a single fraction to confirm it matches the original.

3. Binomial Expansion for Negative and Fractional Powers★★★★☆⏱ 20 min

📘 Definition

General Binomial Expansion

(1 + y)^n

For any real number $n$ and $∣ y ∣ < 1$ , the infinite binomial expansion is: $1 + n y + \frac{n ( n - 1 )}{2 !} y^{2} + \frac{n ( n - 1 ) ( n - 2 )}{3 !} y^{3} + ...$

Example:

Valid for all non-positive-integer $n$ , unlike the finite expansion for positive integers

To expand expressions of the form $(a + b x)^{n}$ where $a > 0$ , first factor out $a^{n}$ to get the standard $(1 + y)^{n}$ form, then apply the general expansion. Always state the range of validity for the expansion.

📐 Worked Example

Find the first three non-zero terms of the expansion of $\frac{1}{1 + 2 x}$ and state its range of validity.

1
Rewrite the function to match the standard form: $(1 + 2 x)^{- 1}$ , so $n = - 1$ , $y = 2 x$
2
Substitute into the general expansion up to the $x^{2}$ term:
3
$(1 + y)^n = 1 + ny + \frac{n(n-1)}{2!}y^2 + ...$
4
$(1 + 2x)^{-1} = 1 + (-1)(2x) + \frac{(-1)(-2)}{2}(2x)^2 + ... = 1 - 2x + 4x^2 - ...$
5
Calculate the range of validity from $∣ y ∣ < 1$ :
6
$|2x| < 1 \implies |x| < \frac{1}{2}$

4. Common Pitfalls

Wrong move:

Skipping polynomial division for improper rational functions before partial fractions

Why:

Partial fraction rules only apply to proper fractions, leading to an incorrect decomposition

Correct move:

Always compare the degrees of numerator and denominator first; divide if $de g (P) \geq de g (Q)$

Wrong move:

Only including the highest degree term for repeated linear factors

Why:

Missing lower degree terms changes the overall expression, leading to wrong constants

Correct move:

For $(a x + b)^{k}$ , include terms from $\frac{A _{1}}{a x + b}$ up to $\frac{A _{k}}{( a x + b ) ^{k}}$

Wrong move:

Forgetting to factor out the leading constant when expanding $(a + b x)^{n}$

Why:

The standard expansion is only valid for $(1 + y)^{n}$ , so all coefficients will be wrong

Correct move:

Factor out $a^{n}$ first to get $a^{n} (1 + \frac{b x}{a})^{n}$ before expanding

Wrong move:

Omitting or incorrectly writing the range of validity for binomial expansion

Why:

Examiners consistently award at least one mark for the correct validity, which is required for all non-positive $n$

Correct move:

Always rearrange $∣ y ∣ < 1$ to get a condition on $x$ and write it clearly

5. Quick Reference Cheatsheet

Concept	Rule
Proper rational function	$de g (P) < de g (Q)$
Improper rational function	$de g (P) \geq de g (Q)$ ; divide first
Distinct linear $(a x + b)$	Term: $\frac{A}{a x + b}$
Repeated linear $(a x + b)^{2}$	Terms: $\frac{A}{a x + b} + \frac{B}{( a x + b ) ^{2}}$
Irreducible quadratic $(a x^{2} + b x + c)$	Term: $\frac{A x + B}{a x ^{2} + b x + c}$
Binomial expansion $(1 + y)^{n}$	$1 + n y + \frac{n ( n - 1 )}{2 !} y^{2} + ...$
Expansion validity	$∣ y ∣ < 1$ for all $n \neq = +$ ve integer

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Partial fraction decomposition
2023 · 2
Binomial expansion of rational function
2021 · 1
Improper fraction simplification

Going deeper

syllabusCIE 9709 A-Level Mathematics Syllabus

What's Next

This module lays the foundation for many core topics in CIE A-Level Pure Mathematics 3. Partial fraction decomposition is a required pre-processing step for integrating rational functions and solving differential equations with separable or linear forms. Binomial expansion of rational functions with negative and fractional powers is used for approximating functions, finding series expansions, and working with infinite sequences. Mastery of these algebraic manipulation skills is essential to access full marks on multi-step questions that combine multiple topics, which are common in P3 exams. Building on these skills will let you tackle more advanced concepts confidently.