O-Level Elementary Mathematics: Algebraic Manipulation & Factorisation – The Core Skills You Must Master

Why Algebraic Manipulation Matters for O-Level E-Math

Algebra is the language of secondary school mathematics. In O-Level Elementary Mathematics, it appears in topics such as equations, inequalities, graphs, functions, and even mensuration and trigonometry. If your algebra is weak, almost every question becomes harder.

This guide focuses on two core skills:

• Algebraic manipulation – simplifying, expanding, rearranging.
• Factorisation – expressing expressions as products of factors.

Strengthening these basics will immediately improve your accuracy and speed across the paper.

Core Algebraic Manipulation Skills

1. Handling Brackets and Like Terms

Many careless mistakes come from rushing through expansion and collection of like terms.

Key ideas

• Distributive law: a(b + c) = ab + ac
• Like terms must have the same variable(s) and power(s).

Example 1

Simplify: 3(2x − 5) − 4(x + 1)

Step-by-step:

1. Expand: 3(2x − 5) = 6x − 15; −4(x + 1) = −4x − 4
2. Combine: (6x − 15) + (−4x − 4) = (6x − 4x) + (−15 − 4) = 2x − 19

Common mistake: Forgetting to multiply the negative sign into the second bracket, leading to −4x + 1 instead of −4x − 4.

2. Working with Fractions and Algebraic Denominators

Algebraic fractions often appear in simplification and equation questions.

Key ideas

• Find the common denominator.
• Only cancel factors, not terms.

Example 2

Simplify: \( \dfrac{3}{x} + \dfrac{2}{x+1} \)

Step-by-step:

1. Common denominator: x(x + 1)
2. Rewrite: \( \dfrac{3(x+1) + 2x}{x(x+1)} = \dfrac{3x + 3 + 2x}{x(x+1)} = \dfrac{5x + 3}{x(x+1)} \)

Common mistake: Adding denominators directly: \( \dfrac{3}{x} + \dfrac{2}{x+1} = \dfrac{5}{2x+1} \) (incorrect).

3. Rearranging Formulae

Rearranging formulae is tested directly and also appears in Physics and Chemistry. Treat the subject you want as the unknown, and perform the same operation on both sides.

Example 3

Make x the subject: \( y = \dfrac{3x - 2}{5} \)

1. Multiply both sides by 5: 5y = 3x − 2
2. Add 2: 5y + 2 = 3x
3. Divide by 3: \( x = \dfrac{5y + 2}{3} \)

Exam tip: Write each step clearly on a new line; this reduces sign errors and makes it easier to check.

Factorisation: The Three Main Types

Most O-Level E-Math factorisation questions fall into three categories. Train yourself to recognise which pattern is being used.

1. Common Factor

This is the simplest type.

Example 4

Factorise: 6x²y − 9xy

Step-by-step:

1. Identify common factors: 3xy
2. Factor out: 6x²y − 9xy = 3xy(2x − 3)

Check: Expand 3xy(2x − 3) to confirm you get back the original expression.

2. Grouping

Used when there are four terms and no obvious single common factor.

Example 5

Factorise: ax + ay + bx + by

Step-by-step:

1. Group terms: (ax + ay) + (bx + by)
2. Factor each group: a(x + y) + b(x + y)
3. Factor out (x + y): (x + y)(a + b)

Exam tip: Rearranging terms is allowed. Try different groupings if the first attempt does not work.

3. Quadratic Trinomials

These are of the form ax² + bx + c. There are two main cases.

Case A: Coefficient of x² is 1

Form: x² + bx + c

Example 6

Factorise: x² + 7x + 12

1. Find two numbers that multiply to 12 and add to 7: 3 and 4
2. Write: (x + 3)(x + 4)

Case B: Coefficient of x² is not 1

Form: ax² + bx + c, a ≠ 1

Example 7

Factorise: 6x² + 11x + 3

1. Multiply a and c: 6 × 3 = 18
2. Find two numbers that multiply to 18 and add to 11: 2 and 9
3. Split middle term: 6x² + 2x + 9x + 3
4. Group: (6x² + 2x) + (9x + 3)
5. Factor each group: 2x(3x + 1) + 3(3x + 1)
6. Factor common bracket: (3x + 1)(2x + 3)

Common mistake: Stopping at 6x² + 11x + 3 = (6x + ? )(x + ? ) without checking the cross terms.

Linking Factorisation to Equations and Graphs

1. Solving Quadratic Equations by Factorisation

Once you can factorise, solving equations becomes straightforward.

Example 8

Solve: x² − 5x + 6 = 0

1. Factorise: x² − 5x + 6 = (x − 2)(x − 3)
2. Set each factor to zero: x − 2 = 0 or x − 3 = 0
3. Solutions: x = 2 or x = 3

Exam tip: Always write both factors = 0. Do not skip directly to the answers.

2. Intercepts of Quadratic Graphs

For y = ax² + bx + c:

• The x-intercepts are the solutions of ax² + bx + c = 0.
• If you can factorise the quadratic, you can find the intercepts quickly.

Example 9

Given y = x² − x − 6, find the x-intercepts.

1. Set y = 0: x² − x − 6 = 0
2. Factorise: (x − 3)(x + 2) = 0
3. Solutions: x = 3, x = −2

So the graph cuts the x-axis at (3, 0) and (−2, 0).

Typical Exam Traps and How to Avoid Them

1. Cancelling Terms Instead of Factors

Wrong:

\( \dfrac{x^2 - 4}{x - 2} = \dfrac{\cancel{x^2} - 4}{\cancel{x} - 2} \) (incorrect)

Correct:

First factorise: x² − 4 = (x − 2)(x + 2)

Then cancel the common factor (x − 2):

\( \dfrac{(x - 2)(x + 2)}{x - 2} = x + 2 \), for x ≠ 2.

2. Dropping Brackets After Substitution

When substituting negative values, always use brackets.

Example 10

Evaluate: 2x² − 3x when x = −4

Correct: 2(−4)² − 3(−4) = 2(16) + 12 = 44

Common mistake: Writing 2−4² − 3−4 and misreading the signs.

3. Forgetting Restrictions on Variables

For expressions with denominators, some values of x are not allowed.

Example 11

Given \( \dfrac{5}{x - 3} \), x ≠ 3 because the denominator cannot be zero.

In some questions, you must state these restrictions in your final answer.

How to Practise Algebra Effectively

1. Build Speed with Short Daily Drills

• Spend 10–15 minutes daily on pure manipulation and factorisation.
• Mix question types: common factor, grouping, quadratics, fractions.

Consistency is more important than doing a long session once a week.

2. Track Your Personal Error Patterns

After each practice, note down:

• Where you lost marks (signs, expansion, factorisation choice).
• The exact step where the error occurred.

Review this list before your next practice to remind yourself what to watch out for.

3. Connect Algebra to Other Topics

When doing questions from other chapters (e.g. graphs, simultaneous equations, inequalities), pay attention to where algebraic manipulation is required. Circle those steps. This helps you see how important algebra is across the paper.

Summary: What You Should Be Able to Do

By the time you sit for O-Level E-Math, you should confidently:

• Simplify and expand expressions with brackets and fractions.
• Factorise using common factor, grouping, and quadratic patterns.
• Solve quadratic equations by factorisation.
• Manipulate formulae and handle algebraic denominators without careless errors.

If any of these skills still feel shaky, focus your revision there first. Strong algebraic manipulation and factorisation will make the rest of E-Math much more manageable.