What this topic covers
Algebra has 22 subtopics, including:
- The basic processes of algebra
- Definition of a function
- Domain and range of a function
- Composite functions
- Inverse functions
- Expanding brackets and collecting like terms
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GCSE AQA Higher Further Maths · Topic 2 of 6
Algebra – GCSE AQA Higher Further Maths revision notes
Choose a subtopic. Algebra covers The basic processes of algebra, Definition of a function, Domain and range of a function, and Composite functions in GCSE AQA Higher Further Maths. 24 lessons in 22 subtopics.
Revise Algebra for GCSE AQA Higher Further Maths, including The basic processes of algebra, Definition of a function, and Domain and range of a function.
24
Lessons
22
Subtopics
Algebra lessons by subtopic
24 lessons, grouped by subtopic.
The basic processes of algebra
- Mastering Basic Algebraic Processes: This lesson covers the fundamental processes of algebra and how to effectively use manipulative algebra skills.
Definition of a function
- Understanding Functions and f(x) Notation: Learn the definition of a function and how to use the f(x) notation with examples.
Domain and range of a function
- Introduction to Domain and Range: Learn to define the domain and range of a function and how they are expressed mathematically.
Composite functions
- Understanding Composite Functions: Learn what composite functions are and how they work. We'll cover the notation and order of operations.
Inverse functions
- Understanding Inverse Functions: This lesson introduces the concept of inverse functions, their notation, and the importance of domains to ensure one-to-one functions.
Expanding brackets and collecting like terms
- Expanding Brackets and Collecting Like Terms: Learn to expand single brackets and simplify expressions by collecting like terms. This lesson covers the basics of algebraic manipulation.
Expand (a + b)^n for positive integer n
- Introduction to Binomial Expansion: Learn how to expand (a + b)^n for positive integers n, and discover Pascal's triangle as a shortcut for finding the coefficients.
Factorising
- Introduction to Factorising: Learn the basics of factorising algebraic expressions.
Manipulation of rational expressions
- Manipulating Rational Expressions: Learn to simplify, add, subtract, multiply, and divide algebraic fractions with numeric, linear, and quadratic denominators.
Use and manipulation of formulae and expressions
- Formulae and Expressions: Use and Manipulation: Learn how to use and manipulate formulae and expressions, including rearranging and simplifying them.
Use of the factor theorem for rational values of the variable for polynomials
- Factor Theorem with Rational Roots: Learn how to use the factor theorem to find rational roots of polynomial equations.
Completing the square
- Introduction to Completing the Square: Learn the basic technique of completing the square for quadratic expressions. Understand how to rewrite a quadratic in the form (x + a)^2 + b.
Drawing and sketching of functions; interpretation of graphs
- Sketching Functions: Linear and Quadratic: Learn to sketch linear and quadratic functions and interpret their graphs.
- Sketching Exponential Functions: Learn to sketch exponential functions of the form y = ab^x and y = ab^-x and interpret their graphs.
Solution of linear and quadratic equations
- Solving Linear and Quadratic Equations: Learn to solve linear and quadratic equations using various methods.
- Quadratic Equations in Context: Apply your knowledge to solve contextual problems that result in linear or quadratic equations.
Algebraic and graphical solution of simultaneous equations in two unknowns
- Solving Simultaneous Equations Algebraically and Graphically: Learn to solve simultaneous equations involving two unknowns using both algebraic and graphical methods. Includes linear and second-order equations.
Algebraic solution of linear equations in three unknowns
- Solving Linear Equations with Three Unknowns: Learn how to solve linear equations involving three unknowns using algebraic methods. We will cover substitution and elimination techniques.
Solution of linear and quadratic inequalities
- Solving Linear and Quadratic Inequalities: Learn how to solve linear and quadratic inequalities, expressing solutions using various notations.
Index laws, including fractional and negative indices and the solution of equations
- Mastering Index Laws: Explore and apply index laws, including fractional and negative indices, to solve equations.
Algebraic proof
- Introduction to Algebraic Proof: Learn the basics of constructing algebraic proofs to demonstrate mathematical statements.
Using nth terms of sequences; limiting value of a sequence as n tends to infinity
- Nth Terms and Limits of Sequences: Explore how to use the nth term to find any term in a sequence and investigate the limiting value of a sequence as n approaches infinity.
nth terms of linear sequences
- Finding the nth Term of Linear Sequences: Learn how to determine the nth term rule for linear sequences. This lesson covers identifying the common difference and using it to build the nth term expression.
nth terms of quadratic sequences
- Finding the nth Term of Quadratic Sequences: Learn how to determine the nth term rule for quadratic sequences.
More in this course
Go back to the full GCSE AQA Higher Further Maths topic list, or practise this topic with past-paper questions.
Frequently asked questions
What is Algebra?
Algebra covers The basic processes of algebra, Definition of a function, Domain and range of a function, and Composite functions in GCSE AQA Higher Further Maths. 24 lessons in 22 subtopics.
How do you expand (a + b)ⁿ?
To expand (a + b)ⁿ, take the coefficients from row n of Pascal's triangle, then write each term with a decreasing power of a and an increasing power of b. The Introduction to Binomial Expansion notes cover this in full.
What are the laws of indices?
The index laws for the same base are: when multiplying, add the indices; when dividing, subtract the indices; when raising a power to a power, multiply the indices. The Mastering Index Laws notes cover this in full.
How do you complete the square?
To complete the square on x² + bx + c, halve the coefficient of x, write the squared bracket, subtract the square of that half, then add the original constant. The Introduction to Completing the Square notes cover this in full.
What is the difference of two squares?
The difference of two squares is the pattern a² − b² = (a + b)(a − b), used when two perfect square terms are separated by a minus sign and nothing else. The Introduction to Factorising notes cover this in full.
How do you find the nth term of a linear sequence?
The nth term of a linear sequence is found by working out the common difference, which becomes the coefficient of n, then finding the constant by subtracting it from the first term. The Finding the nth Term of Linear Sequences notes cover this in full.
How do you show an inequality on a number line?
On a number line, use an open circle at a boundary value that is not part of the solution and a filled circle at one that is (≤ or ≥), then shade a line or arrow in the direction of the solution set. The Solving Linear and Quadratic Inequalities notes cover this in full.
How do you find the nth term of a quadratic sequence?
The nth term of a quadratic sequence is found by halving the second difference to get a, the coefficient of n², subtracting an² from each term, then finding the nth term of the remainder. The Finding the nth Term of Quadratic Sequences notes cover this in full.
How do you factorise a quadratic?
To factorise a quadratic x² + bx + c, find two numbers that multiply to give c and add to give b, then put them in the brackets (x + p)(x + q). The Introduction to Factorising notes cover this in full.