Edexcel A-Level Maths: Algebra and Functions — Complete Revision Guide (9MA0)
Edexcel A-Level Maths: Algebra and Functions — Complete Revision Guide (9MA0)
Algebra and functions is the foundation of Edexcel A-Level Maths (9MA0). It is the second major content area in the specification, sitting just after proof, and it underpins almost every other topic you will study. Calculus depends on confident algebraic manipulation. Coordinate geometry uses quadratic and simultaneous-equation work constantly. Trigonometric proofs rely on the same factorising and partial-fraction techniques. Even the applied papers — mechanics and statistics — assume that algebra is automatic and accurate. If your algebra is shaky, every other topic gets harder. If it is strong, the rest of the course becomes far more approachable.
This guide is a topic-by-topic walkthrough of the algebra and functions content in the 9MA0 specification. It covers everything Edexcel can examine in this section: surds and indices, quadratic functions, simultaneous equations, inequalities, polynomials, the factor and remainder theorems, algebraic fractions, partial fractions, modulus functions, and graph transformations. For each topic you will see the core skills, the typical pitfalls, a short worked example or sketch, and a link to the full lesson on the LearningBro course.
The aim is not to replace working through problems. The only way to get good at algebra is to do hundreds of questions and feel the patterns in your fingers. The aim is to give you a clear map of what you need to know, in the order Edexcel teaches it, so your revision is targeted rather than scattered. Use this guide as a checklist, a refresher, and a launchpad into focused practice.
What the Edexcel 9MA0 Specification Covers
The Edexcel A-Level Maths qualification (9MA0) is assessed through three two-hour papers, each worth 100 marks. Papers 1 and 2 cover pure mathematics, and Paper 3 covers statistics and mechanics. The algebra and functions content sits in Section 2 of the pure specification and can appear on either Paper 1 or Paper 2. There is no choice of questions and no coursework, so every mark must be earned in the exam.
Algebra and functions is one of the highest-frequency topic areas on the pure papers. Almost no question stands entirely outside of it — even a calculus or trigonometry problem will usually require you to factorise, simplify, or rearrange at some point. The table below shows the sub-topics of Section 2, the part of the specification they sit under, and a realistic estimate of how many marks across a Paper 1 / Paper 2 sitting come from each.
| Topic | Spec Section | Typical Paper 1 marks weight |
|---|---|---|
| Surds, indices and rational powers | 2.1 | 2-4 marks |
| Quadratic functions | 2.2 | 6-10 marks |
| Simultaneous equations | 2.3 | 4-6 marks |
| Inequalities | 2.4 | 4-6 marks |
| Polynomials | 2.5 | 4-6 marks |
| Factor and remainder theorems | 2.6 | 4-6 marks |
| Algebraic fractions | 2.7 | 3-5 marks |
| Partial fractions | 2.8 | 3-5 marks |
| Modulus functions | 2.9 | 4-6 marks |
| Graph transformations | 2.10 | 4-6 marks |
These weights are estimates based on the spread of typical 9MA0 papers — not guarantees for any single year. What is reliable, however, is that algebra and functions is consistently the most heavily examined area of pure on Paper 1, and that the same skills resurface inside calculus and trigonometry questions on Paper 2. Mastering this section is high-leverage revision.
Surds, Indices and Rational Powers
Surds and indices are the first topic in the specification because every later topic assumes them. A surd is an irrational root left in exact form, such as sqrt(2) or 3*sqrt(5). Indices are powers, including fractional and negative ones, governed by the index laws. You must be completely fluent with both before tackling anything else.
The core skills are: simplifying surds (sqrt(50) = 5*sqrt(2)), adding and subtracting like surds, multiplying surds, and rationalising the denominator — multiplying top and bottom of a fraction by the conjugate to clear surds from the denominator. For indices you need the laws x^a * x^b = x^(a+b), x^a / x^b = x^(a-b), (x^a)^b = x^(ab), x^0 = 1, x^(-n) = 1/x^n, and x^(1/n) = nth root of x. Combinations like x^(3/2) = sqrt(x^3) appear constantly.
Many candidates lose marks on surds questions through small errors. A common pitfall is treating sqrt(a + b) as sqrt(a) + sqrt(b) — it is not. Another is forgetting to multiply both numerator and denominator by the conjugate when rationalising. With indices, students often misread x^(-1/2) as -sqrt(x) instead of 1/sqrt(x), or apply the power rule before the chain of operations is fully sorted out.
A short worked example. Rationalise (3 + sqrt(2)) / (1 - sqrt(2)). Multiply top and bottom by the conjugate (1 + sqrt(2)): the denominator becomes 1 - 2 = -1, and the numerator becomes 3 + 3sqrt(2) + sqrt(2) + 2 = 5 + 4sqrt(2). The answer is therefore -(5 + 4sqrt(2)) = -5 - 4sqrt(2). Notice how the conjugate clears the surd cleanly because (a - b)(a + b) = a^2 - b^2.
For full coverage with practice questions and worked solutions, see the Surds and Indices lesson.
Quadratic Functions
Quadratics dominate algebra at A-Level. A quadratic function has the general form f(x) = ax^2 + bx + c, where a is non-zero. Edexcel expects you to be fluent with three different forms of the same quadratic, each useful for a different purpose, and to switch between them at speed.
The general form ax^2 + bx + c is convenient for substituting values and for the quadratic formula. The factorised form a(x - p)(x - q) immediately gives you the roots p and q. The completed-square form a(x - h)^2 + k immediately gives you the vertex (h, k) and tells you whether the parabola has a minimum (a > 0) or maximum (a < 0). Sketching a quadratic without first putting it into completed-square form is doing yourself a disservice.
The discriminant b^2 - 4ac tells you about the roots without solving the equation. If b^2 - 4ac > 0 there are two distinct real roots; if b^2 - 4ac = 0 there is a repeated root (the parabola touches the x-axis); and if b^2 - 4ac < 0 there are no real roots (the parabola sits entirely above or below the x-axis). Discriminant questions are extremely common — often phrased as "find the values of k for which the equation has real roots" — and reduce to solving an inequality in k.
A common pitfall is forgetting that "no real roots" still gives a valid graph: the curve simply does not cross the x-axis. Another is sign errors when completing the square with negative coefficients. To complete the square on x^2 - 6x + 11, take half of -6 to get -3, square it to get 9, and write x^2 - 6x + 11 = (x - 3)^2 - 9 + 11 = (x - 3)^2 + 2. The vertex is (3, 2) and the curve has a minimum value of 2.
For the full topic with sketching practice and discriminant problems, see the Quadratic Functions lesson.
Simultaneous Equations
Edexcel tests two flavours of simultaneous equations: pairs of linear equations, and linear-quadratic systems. The linear pair is GCSE-level revision, but the linear-quadratic system is a genuine A-Level skill and a frequent source of marks.
The standard method for a linear-quadratic pair is substitution. Rearrange the linear equation to make x or y the subject, substitute into the quadratic, and solve. You will end up with a quadratic in one variable, which factorises (or yields to the quadratic formula) to give the x-values of the intersection points. Substitute back into the linear equation to find the corresponding y-values.
The discriminant has a powerful geometric meaning here. If a line and a curve are simultaneous and you reduce them to a quadratic, the discriminant of that quadratic tells you about their intersection. Two distinct real roots means the line cuts the curve at two points. A repeated root means the line is tangent to the curve. No real roots means the line and the curve do not meet. Tangency conditions are a classic Edexcel question — you are given a parameter k inside one of the equations and asked to find the value of k for which the line is tangent to the curve.
A common pitfall is solving for x and forgetting to find the corresponding y-values, or finding only one intersection when the question implicitly asks for two. Another is making algebraic errors when the substitution produces a messy expansion. Take your time with the algebra; rushing here costs more marks than rushing anywhere else.
For worked tangency examples and a clean substitution method, see the Simultaneous Equations lesson.
Inequalities
Inequalities at A-Level extend GCSE work into three connected areas: linear inequalities, quadratic inequalities, and rational inequalities. The mechanics of solving look similar to equations, but the reasoning is different and a single careless step can flip the answer.
For linear inequalities, the only rule that matters beyond standard algebra is that multiplying or dividing both sides by a negative number reverses the inequality sign. This is the single most common source of lost marks at A-Level. Many candidates know the rule but forget it under exam pressure, especially when the negative is hidden inside a longer manipulation.
For quadratic inequalities, the right method is graphical. Rearrange so one side is zero, find the roots of the corresponding quadratic, sketch the parabola, and read off the region where the curve sits above or below the x-axis as required. Algebraic methods like dividing through by an unknown sign are unreliable and usually wrong.
For rational inequalities of the form (x + 1) / (x - 2) > 0, do not multiply both sides by (x - 2) because its sign is unknown. Instead, find the critical values where numerator or denominator equals zero, draw a number line, and test the sign of the expression in each interval. The solution is the union of intervals where the expression has the correct sign. Be careful with strict versus non-strict inequalities at the critical values.
For sketching diagrams and a clean sign-table method, see the Inequalities lesson.
Polynomials
A polynomial is an expression of the form a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where the powers are non-negative integers and the coefficients are real. The degree is the highest power of x. Edexcel expects you to recognise polynomials of degree up to 4 and to reason about their behaviour from the coefficients.
The end behaviour of a polynomial is determined by its leading term. For a positive leading coefficient and even degree, the curve rises on both sides; for a positive leading coefficient and odd degree, it falls on the left and rises on the right. Negative leading coefficients flip the picture. This single observation lets you sketch any polynomial confidently if you know its roots.
Multiplicity is the second sketching tool. If a root has multiplicity 1, the curve crosses the x-axis cleanly. If it has multiplicity 2, the curve touches the x-axis and turns back. If it has multiplicity 3, the curve flattens through the x-axis. So a polynomial like y = (x - 1)^2 (x + 2) has a double root at x = 1 (curve touches) and a simple root at x = -2 (curve crosses), with a positive leading coefficient that rises on the right.
A common pitfall is treating a repeated root as two distinct intersections. Another is sketching the curve to scale rather than to shape — for an A-Level sketch, accurate proportions are not required, but the key features (intercepts, end behaviour, turning behaviour at multiple roots) must be correct.
For sketching practice and a workflow for any polynomial, see the Polynomials lesson.
Factor and Remainder Theorems
The factor theorem says: (x - a) is a factor of a polynomial f(x) if and only if f(a) = 0. The remainder theorem says: when f(x) is divided by (x - a), the remainder is f(a). These two results are quick to state but powerful in practice — they are the standard tools for factorising cubics and higher-order polynomials.
The workflow for factorising a cubic f(x) = x^3 + bx^2 + cx + d is to test small integer values until you find one that gives f(a) = 0, then factor out (x - a) by polynomial division (or inspection), leaving a quadratic factor. The quadratic factor is then handled by the standard quadratic methods. The candidates to test are the divisors of the constant term d — by the rational root theorem, any rational root must divide d.
A short worked example. Factorise f(x) = x^3 - 2x^2 - 5x + 6. Test x = 1: f(1) = 1 - 2 - 5 + 6 = 0. So (x - 1) is a factor. Dividing through gives x^3 - 2x^2 - 5x + 6 = (x - 1)(x^2 - x - 6) = (x - 1)(x - 3)(x + 2). The roots are x = 1, 3, -2.
Common pitfalls include sign errors during polynomial division, particularly when the cubic has negative coefficients, and failing to check enough small values before moving to the formula. Test x = 1, -1, 2, -2 first; one of them is usually a root in exam questions.
For polynomial division technique and worked cubic factorisations, see the Factor Theorem lesson.
Algebraic Fractions
Algebraic fractions are fractions whose numerator and denominator are polynomials. Working with them requires nothing more than the standard fraction rules from arithmetic — but the algebra inside makes them feel harder than they are.
The four core operations are simplifying, adding/subtracting, multiplying, and dividing. To simplify, factorise numerator and denominator and cancel common factors. Cancellation is only valid for factors, not terms — a frequent source of errors. To add or subtract algebraic fractions, find a common denominator (usually the product of the two denominators, simplified by any shared factors), rewrite each fraction, and combine numerators. To multiply, factorise everything, cancel diagonally, and write the answer as a single fraction. To divide, multiply by the reciprocal.
A separate skill is converting between improper and mixed algebraic fractions. An improper algebraic fraction is one where the degree of the numerator is at least the degree of the denominator. To convert it to mixed form (a polynomial plus a proper algebraic fraction), do polynomial division. This is essential preparation for partial fractions and for some integration techniques later in the course.
The most common pitfall is cancelling terms instead of factors. The expression (x + 2) / (x + 4) cannot be simplified to 2 / 4 — the +2 and +4 are not factors, they are parts of a sum. You can only cancel a complete bracketed factor that appears in both numerator and denominator.
For step-by-step practice on each of the four operations, see the Algebraic Fractions lesson.
Partial Fractions
Partial fractions is the reverse of adding algebraic fractions: starting from a single fraction with a complicated denominator, you split it into a sum of simpler fractions. Edexcel uses partial fractions in three places — algebra (this section), the binomial expansion of (1 + x)^n where n is not a positive integer, and integration. So the technique is high-value across the course.
There are three cases on the 9MA0 specification, distinguished by what is in the denominator after factorising.
| Case | Denominator factor | Partial fraction form |
|---|---|---|
| Distinct linear factors | (ax + b)(cx + d) | A/(ax+b) + B/(cx+d) |
| Repeated linear factor | (ax + b)^2 | A/(ax+b) + B/(ax+b)^2 |
| Irreducible quadratic factor | (x^2 + px + q) where the discriminant is negative | (Ax+B)/(x^2+px+q) |
In each case, write the original fraction as a sum of unknown-coefficient fractions of the right form, multiply both sides by the original denominator, and solve for the unknowns. There are two standard methods for solving: substituting strategic values of x to make terms vanish (the cover-up method for distinct linear factors), and equating coefficients of like powers.
The cover-up method is fast for distinct linear factors. To find A in A/(x - 1) + B/(x + 2) = (3x + 4) / ((x - 1)(x + 2)), cover the (x - 1) on the right and substitute x = 1 into what remains: A = (3 + 4) / (1 + 2) = 7/3. Repeat for B by covering (x + 2) and substituting x = -2.
A common pitfall is using the wrong form for the case in front of you — for instance, writing (Ax + B)/(x - 1)^2 instead of A/(x - 1) + B/(x - 1)^2 for a repeated linear factor. Always check the case before writing the partial-fraction template.
For worked examples in all three cases and the cover-up method, see the Partial Fractions lesson.
Modulus Functions
The modulus function |x| returns the magnitude of x, ignoring sign. So |3| = 3 and |-3| = 3. More formally, |x| = x when x is non-negative and |x| = -x when x is negative. This piecewise definition is the foundation of every modulus problem.
For sketching, you need to distinguish two transformations carefully. y = |f(x)| takes any part of y = f(x) that lies below the x-axis and reflects it upward. y = f(|x|) takes the part of y = f(x) for x non-negative and reflects it across the y-axis to produce a graph symmetric about the y-axis. The two transformations look similar at first but produce very different graphs, and exam questions test both.
To solve modulus equations like |2x - 3| = 5, split into cases. Either 2x - 3 = 5, giving x = 4, or 2x - 3 = -5, giving x = -1. Always check the answers in the original equation, because both branches produce candidate solutions but only some satisfy the equation. For modulus equations with a modulus on each side, like |x + 2| = |2x - 1|, square both sides to remove the modulus signs and solve the resulting quadratic.
A common pitfall is forgetting that a modulus equation can have zero, one, or two solutions depending on the right-hand side. If the right-hand side is negative, the equation has no solution because |...| is always non-negative. Many candidates lose marks by writing two solutions on autopilot rather than checking validity.
For sketching practice on both transformations and a clean case-split workflow for equations, see the Modulus Functions lesson.
Graph Transformations
Graph transformations are the family of operations that move, stretch, or reflect the graph y = f(x) to produce related graphs. Edexcel expects you to know all the standard transformations and to apply them in any order, including to unfamiliar functions where you cannot see the algebra.
| Transformation | Effect on the graph |
|---|---|
| y = f(x) + a | Translation by a units in the positive y-direction |
| y = f(x + a) | Translation by a units in the negative x-direction |
| y = a*f(x) | Vertical stretch by scale factor a |
| y = f(ax) | Horizontal stretch by scale factor 1/a |
| y = -f(x) | Reflection in the x-axis |
| y = f(-x) | Reflection in the y-axis |
The single rule that catches most candidates is the "inside-bracket do the opposite" rule. Anything that happens to x inside the bracket affects the graph in the opposite way to what you might expect. Adding 3 inside the bracket — y = f(x + 3) — translates the graph 3 units to the left, not the right. Multiplying x by 2 inside the bracket — y = f(2x) — stretches the graph by a factor of 1/2 in the x-direction, not 2. Outside-the-bracket transformations behave intuitively; inside-the-bracket transformations invert.
For combining transformations, the order matters and follows the order of operations applied to the input. For y = 2*f(x + 1) - 3 starting from y = f(x): first replace x by x + 1 (translate left by 1), then multiply by 2 (vertical stretch by 2), then subtract 3 (translate down by 3). Drawing the intermediate graphs is far safer than trying to do it all at once.
A common pitfall is mixing up the direction of horizontal translations or stretches. Another is applying transformations in the wrong order — for example, doing the vertical translation before the vertical stretch when the algebra demands the reverse.
For step-by-step transformation sketches and combined transformations, see the Transformations lesson.
Common Mark-Loss Patterns Across Algebra and Functions
Across the whole algebra and functions section, a small set of habits accounts for a disproportionate share of lost marks. None of these are about content you do not know. They are all about content you do know, applied carelessly.
- Sign errors when expanding negative brackets. -3(x - 2) is -3x + 6, not -3x - 6. Many candidates lose marks here in the final line of an otherwise correct piece of work.
- Cancelling terms instead of factors. A common pitfall is writing (x + 4) / (x + 2) as 2 — you can only cancel complete bracketed factors that appear in both numerator and denominator.
- Forgetting to flip the inequality when multiplying or dividing by a negative. Slowing down and underlining each negative-multiplication step is a cheap and effective discipline.
- Solving a quadratic and not finding both intersection coordinates in simultaneous-equation problems. The question usually asks for points; an x-value alone is half the answer.
- Sketching to scale rather than to shape. Examiners want intercepts, end behaviour, and turning points correct. They do not want a perfectly proportioned graph.
- Using the wrong partial-fraction template for repeated factors or irreducible quadratics. Always check the case before assigning unknown coefficients.
- Treating modulus equations as automatically two-solution problems without checking which branches actually satisfy the original equation.
- Mis-applying the inside-bracket rule for graph transformations. Inside the bracket, do the opposite.
- Not showing enough working. Edexcel mark schemes award method marks generously when the working is clear. A correct final answer with no working can score fewer marks than an incorrect final answer with clean method.
Many candidates lose marks here every series. A revision plan that explicitly drills these habits — not just the content — will move your grade more than another pass through the textbook.
Recommended Six-Week Revision Plan
This plan is designed for a candidate who has covered the algebra and functions content in lessons but wants to revise it cleanly before the exam. It assumes about 5-6 hours per week on this section. Adjust pace if you are starting earlier or later.
| Week | Topics | Practice |
|---|---|---|
| 1 | Surds, indices, rational powers; quadratic functions (all three forms, discriminant) | 30-40 short questions on surds and indices; 15 quadratic-form-conversion problems; 10 discriminant problems |
| 2 | Simultaneous equations (linear-linear and linear-quadratic); inequalities (linear, quadratic, rational) | 15 simultaneous-equation problems including 5 with tangency; 20 inequality problems including 5 rational |
| 3 | Polynomials; factor and remainder theorems | 10 polynomial-sketching problems; 15 cubic-factorisation problems using the factor theorem |
| 4 | Algebraic fractions; partial fractions (all three cases) | 20 algebraic-fraction operations; 15 partial-fraction decompositions covering each case |
| 5 | Modulus functions; graph transformations (single and combined) | 10 modulus-equation problems; 10 modulus-sketching problems; 15 transformation problems |
| 6 | Mixed practice; targeted review of weakest topics; full algebra-and-functions question sets | One full mixed problem set per day; review marking-scheme working for any question scoring below 60% |
The point of the plan is to keep moving forward while maintaining contact with earlier topics. Do not spend three weeks on quadratics and run out of time before partial fractions. By the end of week 5, every topic in the section should have had focused contact and a practice round. Week 6 is consolidation and weakness-targeting.
A useful discipline through the whole plan is to treat any question you got wrong not as a mistake but as a diagnostic. Was it a content gap? A method error? A careless arithmetic slip? Logging the cause means your next review session targets the right thing.
How LearningBro's Edexcel A-Level Maths Algebra Course Helps
LearningBro's Edexcel A-Level Maths: Algebra and Functions course is built around the structure of this guide. Each of the ten lessons covers one section of the 9MA0 specification, in the order Edexcel teaches it, with worked examples, practice questions and full mark-scheme-style solutions. Lessons end with a short review and quick-recall questions designed for spaced revisits.
The course is designed to be used in two ways. As a first pass, you can work through the lessons in order, building each topic on the last. As a revision tool, you can drop into any lesson and work the practice independently — for example, drilling partial fractions for a week before mocks. The AI tutor is available throughout to give targeted hints when you get stuck, without giving away full solutions, and to mark your written working with structured feedback.
If you want one place to revise this section of the spec well, with realistic practice and clean explanations of every topic, the full course is the right next step. Start with the Edexcel A-Level Maths: Algebra and Functions course.