How do I solve a quadratic inequality like x^2 - 5x + 6 > 0?

First, factorise the quadratic: (x-2)(x-3) > 0. Then find the critical values where the expression equals zero: x=2 and x=3. Sketch the graph of y = (x-2)(x-3) – it's a U-shaped parabola crossing the x-axis at 2 and 3. The inequality >0 means we want the parts of the graph above the x-axis. This occurs when x 3. In set notation: {x: x 3}. Always check a test point in each region to confirm.

What is the factor theorem and how do I use it?

The factor theorem states that for a polynomial f(x), if f(a)=0 then (x-a) is a factor. To use it, test integer factors of the constant term. For example, for f(x)=x^3-6x^2+11x-6, test x=1: f(1)=1-6+11-6=0, so (x-1) is a factor. Then divide f(x) by (x-1) to get a quadratic, which you can factorise further. This helps factorise cubics and higher-degree polynomials.

When do I use 'and' vs 'or' in inequality solutions?

Use 'and' when the solution must satisfy both inequalities simultaneously – this gives the intersection of the solution sets. For example, x > 2 and x 5 means all numbers less than 2 or greater than 5. In set notation, 'and' is written as an intersection (∩) and 'or' as a union (∪).

How do I simplify rational expressions like (x^2-4)/(x^2+2x)?

Factorise numerator and denominator completely. Numerator: x^2-4 = (x-2)(x+2). Denominator: x^2+2x = x(x+2). Cancel the common factor (x+2), leaving (x-2)/x, provided x ≠ -2 and x ≠ 0 (since division by zero is undefined). Always state any restrictions on the variable after simplifying.

What is algebraic division and when is it used?

Algebraic division is used to divide a polynomial by a linear divisor (e.g., x - c). It is similar to long division with numbers. For example, divide 2x^3 - 3x^2 + 4x - 5 by x - 2. You get a quotient and a remainder. This is useful for factorising polynomials (if remainder is zero, divisor is a factor) and for simplifying rational expressions where the numerator degree is higher than the denominator.

How do I represent inequalities graphically?

For linear inequalities like y > 2x + 1, draw the line y = 2x + 1 (dashed if > or , below for y <. For quadratic inequalities like y ≤ x^2 - 4, draw the parabola y = x^2 - 4 (solid line) and shade the region below it (since y is less than or equal). For systems of inequalities, shade the region that satisfies all conditions – the overlapping region is the solution.

Express solutions through correct use of 'and' and 'or', or through set notation; represent linear and quadratic inequalities graphically; manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem; simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only)

EDEXCEL

A-Level

This topic covers the algebraic manipulation of polynomials, including expansion, factorisation, and division by linear expressions, alongside the application of the factor theorem. It also encompasses the simplification of rational expressions and the graphical representation of linear and quadratic inequalities, including the use of set notation and logical connectors.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Topic Overview

This topic covers the essential algebraic manipulation skills required for Edexcel A-Level Mathematics. You will learn to solve inequalities and represent them using set notation or the words 'and'/'or', as well as graphically. This includes linear and quadratic inequalities, where you must interpret regions on a graph. These skills are fundamental for modelling real-world constraints in mechanics, statistics, and decision mathematics.

Polynomial manipulation is a core skill: expanding brackets, collecting like terms, factorising (including by grouping and using the factor theorem), and performing algebraic division by linear expressions. You will also simplify rational expressions by factorising and cancelling common factors. Mastery of these techniques is crucial for solving equations, analysing functions, and preparing for calculus.

These topics underpin much of the A-Level course. For example, factorising polynomials is essential for sketching graphs, solving equations, and integrating rational functions. Algebraic division and the factor theorem are used to find roots and factorise higher-degree polynomials. Understanding inequalities is vital for domain and range questions, optimisation, and probability. Together, these skills build algebraic fluency that is tested across all papers.

Key Concepts

Core ideas you must understand for this topic

→Solving linear and quadratic inequalities: treat like equations but reverse the inequality sign when multiplying/dividing by a negative. For quadratics, sketch the graph or use a sign diagram.
→Set notation for solutions: use {x: condition} or interval notation. 'And' means intersection (both conditions true), 'or' means union (at least one true).
→Polynomial manipulation: expand brackets using FOIL or the distributive law, collect like terms, factorise using common factors, difference of squares, trinomials, and grouping.
→Factor theorem: if f(a)=0, then (x-a) is a factor. Use this to factorise cubics and higher polynomials by testing integer factors of the constant term.
→Algebraic division: divide a polynomial by a linear divisor (e.g., x - c) using long division or synthetic division. The quotient and remainder can be used to factorise or simplify rational expressions.

What You Need to Demonstrate

Key skills and knowledge for this topic

Correct use of 'and'/'or' and set notation for inequality solutions
Accurate shading and use of dotted/solid lines for graphical inequalities
Correct application of the factor theorem: if f(b/a) = 0, then (ax - b) is a factor
Correct algebraic division of polynomials by linear expressions
Correct factorisation and cancellation of rational expressions
Correct identification of the range of x for which a curve is below a line

Marking Points

Key points examiners look for in your answers

Correct use of 'and'/'or' and set notation for inequality solutions
Accurate shading and use of dotted/solid lines for graphical inequalities
Correct application of the factor theorem: if f(b/a) = 0, then (ax - b) is a factor
Correct algebraic division of polynomials by linear expressions
Correct factorisation and cancellation of rational expressions
Correct identification of the range of x for which a curve is below a line

Examiner Tips

Expert advice for maximising your marks

💡Always check for missing powers of x when performing algebraic division
💡When sketching inequalities, draw the boundary line first as an equation before deciding on the shading
💡Use the factor theorem to test potential roots before attempting full algebraic division
💡Ensure set notation is written in the correct format, e.g., {x : x < a} ∪ {x : x > b}
💡When simplifying rational expressions, factorise both numerator and denominator completely before cancelling
💡When solving quadratic inequalities, always sketch a graph or use a sign diagram to determine the correct region. Marks are often awarded for the correct inequality signs and set notation.
💡In algebraic division, always write the polynomial in descending powers, including zero coefficients for missing terms (e.g., x^3 + 0x^2 - 4x + 2). This avoids sign errors.
💡For rational expressions, factorise numerator and denominator completely before cancelling. Do not cancel terms that are not common factors (e.g., cancelling x in (x+1)/(x+2) is wrong).

Common Mistakes

Pitfalls to avoid in your exam answers

Confusing the use of 'and'/'or' when combining inequality intervals
Incorrectly identifying the factor from the factor theorem (e.g., using (x+b) instead of (ax-b))
Failing to use dotted lines for strict inequalities (< or >) and solid lines for non-strict inequalities (≤ or ≥)
Errors in algebraic division, particularly with missing terms in the polynomial (e.g., forgetting the 0x term)
Cancelling terms incorrectly in rational expressions (e.g., cancelling terms that are added rather than factors)
Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always check: if you multiply both sides of -2x < 6 by -1, you get 2x > -6.
Confusing 'and' with 'or' in compound inequalities. For example, x > 2 and x < 5 means 2 < x < 5 (intersection), while x < 2 or x > 5 means two separate intervals (union).
Thinking that factorising always gives integer factors. The factor theorem may require testing non-integer rational roots (e.g., x = 1/2) if the polynomial has fractional coefficients.

Frequently Asked Questions

Common questions students ask about this topic

Before You Start

Prior knowledge that will help with this topic

•Basic algebra: solving linear equations, expanding single brackets, collecting like terms.
•Quadratic equations: solving by factorising, completing the square, and using the quadratic formula.
•Graphs of linear and quadratic functions: understanding gradient, intercepts, and shape of parabolas.

Key Terminology

Essential terms to know

Polynomial Arithmetic and Algebraic Division
The Factor Theorem and Remainder Theorem Applications
Logical Connectives and Set Notation in Inequalities
Graphical Representation of Regions and Feasible Sets

Likely Command Words

How questions on this topic are typically asked

Solve

Sketch

Factorise

Simplify

Show that

Ready to test yourself?

Practice questions tailored to this topic

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in EDEXCEL A-Level Mathematics

Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations

Apply differentiation to find gradients, tangents and normals; maxima and minima and stationary points; points of inflection; identify where functions are increasing or decreasing

Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form

Topic Synopsis

Key Concepts & Core Principles

Exam Tips & Revision Strategies

Common Misconceptions & Mistakes to Avoid

Examiner Marking Points

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

How do I solve a quadratic inequality like x^2 - 5x + 6 > 0?

What is the factor theorem and how do I use it?

When do I use 'and' vs 'or' in inequality solutions?

How do I simplify rational expressions like (x^2-4)/(x^2+2x)?

What is algebraic division and when is it used?

How do I represent inequalities graphically?

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in EDEXCEL A-Level Mathematics

Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations

Apply differentiation to find gradients, tangents and normals; maxima and minima and stationary points; points of inflection; identify where functions are increasing or decreasing

Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form