AQA GCSE Mathematics (8300) — Algebra
Every time you multiply two linear brackets together, you generate four terms that usually simplify into a quadratic expression. Expansion is the process of removing brackets by multiplying the terms within them. To expand the product of two binomial expressions (expressions containing exactly two terms), the distributive law states that every term in the first bracket must be multiplied by every term in the second.
The FOIL Method is a popular way to ensure all four multiplications are completed: First, Outside, Inside, Last. Once expanded, you must collect like terms, which are terms with the same variable raised to the same power. Note that an expression like does NOT simply equal ; you must rewrite it as before expanding.
Example 1: Expanding Two Binomials Calculate the expansion of .
Higher Tier students must also expand three linear binomials, which results in a cubic expression (where the highest power is ). The commutative law means the order in which you multiply the brackets does not affect the final answer. To do this, multiply the first two brackets, simplify, and then multiply that quadratic by the third bracket.
Example 2: Triple Expansion Calculate the expansion of .
You know how to expand brackets to create a quadratic, but can you run the process in reverse? To factorise an expression is to write it as a product of its factors. Before starting, always check for a Highest Common Factor across all terms, as pulling this out first makes the rest of the calculation much easier.
A monic quadratic has a leading coefficient (the numerical value in front of the ) of 1, written in the form . To factorise these, you must find two numbers that multiply to make the constant (the '' term) and add to make the '' coefficient. The signs of your factors depend on the constant: if is negative, one factor is positive and one is negative.
Example 3: Factorising a Monic Quadratic Calculate the factorisation of .
A non-monic quadratic has a leading coefficient that is not 1 (where ). These are factorised using the "ac" method and splitting the middle term. First, find two numbers that multiply to the product of and , and add to make .
Example 4: Factorising a Non-Monic Quadratic Calculate the factorisation of .
Surprising as it sounds, you can calculate in your head in seconds without a calculator. This relies on the Difference of Two Squares, a specific algebraic identity where . Applying this makes mental maths simple: .
For this identity to work on algebra, the expression must have exactly two terms, both must be perfect squares, and crucially, they must be separated by a minus sign. You can NOT apply this to an expression like because it does not have a minus sign. Any variable with an even power (like ) is a perfect square because .
Sometimes, an AQA exam question will ask you to factorise fully. This command phrase often signals that you must extract a common factor before the difference of two squares becomes visible (for example, ).
Example 5: Difference of Two Squares with Coefficients Calculate the factorisation of .
Example 6: Difference of Two Squares with Even Powers Calculate the factorisation of .
Students often assume (x + y)² = x² + y², but you must rewrite it as (x + y)(x + y) and expand it to ensure you don't miss the middle terms.
Attempting to use the difference of two squares on an expression with a plus sign, like x² + 25; this is impossible because the identity requires a minus sign.
AQA examiners note that students using the visual Grid Method for the final multiplication in a triple expansion are much less likely to miss terms or make sign errors.
If an AQA question asks you to 'Solve... by factorising', remember that factorising is only the first step; you must then set each bracket to zero to find the final values of x.
Always expand your factorised brackets mentally as a quick check—if you don't get your starting expression, you have made a sign or arithmetic error.
When using the difference of two squares on variables with higher powers, remember that taking the square root simply means halving the even power (e.g., the square root of x⁶ is x³).
Binomial
An algebraic expression containing exactly two terms, such as x + 3.
Expansion
The process of removing brackets from an expression by multiplying the terms within them.
Distributive law
The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
FOIL Method
A mnemonic used to remember the steps for expanding two binomials: First, Outside, Inside, Last.
Like terms
Algebraic terms that have the exact same variables raised to the exact same powers.
Quadratic
An algebraic expression where the highest power of the unknown variable is 2, typically in the form ax² + bx + c.
Cubic
An algebraic expression where the highest power of the unknown variable is 3.
Commutative law
The rule that states the order in which terms are multiplied does not affect the final product.
Factorise
The process of writing an expression as a product of its factors, which is the reverse of expanding brackets.
Highest Common Factor
The largest expression or number that divides exactly into two or more terms.
Monic
A quadratic expression where the leading coefficient (the number before x²) is exactly 1.
Coefficient
The numerical value placed before and multiplying a variable in an algebraic expression.
Constant
A standalone number in an expression that does not have a variable attached to it.
Non-monic
A quadratic expression where the leading coefficient (the number before x²) is not equal to 1.
Difference of Two Squares
An algebraic identity stating that a² - b² can be factorised into the two brackets (a + b)(a - b).
Factorise fully
A command phrase indicating that an expression must be broken down completely, often requiring a common factor to be pulled out before using another method like the difference of two squares.
Put your knowledge into practice — try past paper questions for Mathematics
Binomial
An algebraic expression containing exactly two terms, such as x + 3.
Expansion
The process of removing brackets from an expression by multiplying the terms within them.
Distributive law
The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
FOIL Method
A mnemonic used to remember the steps for expanding two binomials: First, Outside, Inside, Last.
Like terms
Algebraic terms that have the exact same variables raised to the exact same powers.
Quadratic
An algebraic expression where the highest power of the unknown variable is 2, typically in the form ax² + bx + c.
Cubic
An algebraic expression where the highest power of the unknown variable is 3.
Commutative law
The rule that states the order in which terms are multiplied does not affect the final product.
Factorise
The process of writing an expression as a product of its factors, which is the reverse of expanding brackets.
Highest Common Factor
The largest expression or number that divides exactly into two or more terms.
Monic
A quadratic expression where the leading coefficient (the number before x²) is exactly 1.
Coefficient
The numerical value placed before and multiplying a variable in an algebraic expression.
Constant
A standalone number in an expression that does not have a variable attached to it.
Non-monic
A quadratic expression where the leading coefficient (the number before x²) is not equal to 1.
Difference of Two Squares
An algebraic identity stating that a² - b² can be factorised into the two brackets (a + b)(a - b).
Factorise fully
A command phrase indicating that an expression must be broken down completely, often requiring a common factor to be pulled out before using another method like the difference of two squares.
