Imagine trying to pack identical sets of items into the fewest possible boxes; finding the highest common factor lets you simplify mathematical expressions in a very similar way. To factorise an expression means to write it as a product of its factors. It is the exact opposite of expanding brackets, working backwards to reverse the distributive law.
When terms share common elements, we simplify the expression into a single bracket. To do this completely, we must identify the Highest Common Factor (HCF) of all terms in the expression. This HCF is placed outside the bracket.
Finding the overall HCF requires looking at two parts of each term:
Factorise fully:
Step 1: Find the HCF of the numbers.
Step 2: Find the HCF of the variables.
Step 3: Combine to find the overall HCF.
Step 4: Divide each term by the HCF to find the contents of the bracket.
Step 5: Write the final single-bracket expression.
Working backwards to find which two numbers were multiplied together is a brilliant puzzle that unlocks quadratic equations. A monic quadratic is an expression where the leading coefficient (the number in front of the ) is exactly 1. When a quadratic expression contains three terms, it is known as a trinomial.
To factorise a monic quadratic trinomial, the result will take the form of double brackets, written as . To find the unknown values and , we use the sum and product method. You must find factor pairs of the constant term () that multiply to give , but add together to give the coefficient of ().
The sign of the constant term gives you a huge clue:
Factorise
Step 1: Identify and .
Step 2: List factor pairs of () and check their sums.
Step 3: Place these numbers into double brackets.
Step 4: Verify by expanding mentally.
When the term has a number in front of it, the simple sum-and-product rule isn't enough; we need a more structured method to break the expression apart. A non-monic quadratic is a quadratic where the leading coefficient () is not 1. To factorise these, we use a systematic process called the ac method.
First, multiply the leading coefficient () by the constant term () to find the "master product". You then find two numbers that multiply to this value and add to give the middle coefficient (). These numbers are used for splitting the middle term, rewriting the original three-term trinomial into four terms.
Once you have four terms, you can use factorising by grouping. You split the expression down the middle, take a common factor from the first pair, and a common factor from the second pair. This will reveal an identical bracket in both parts, which becomes one of your final double brackets.
Factorise
Step 1: Identify , , and , then calculate .
Step 2: Find factors of that sum to .
Step 3: Split the middle term () using these factors.
Step 4: Factorise by grouping the pairs.
Step 5: Write the final factorised expression by combining the outside terms into one bracket, and the common bracket into the other.
When taking out algebraic factors, students often think the Highest Common Factor of x³ and x⁵ is x⁵. It is actually the variable raised to the lowest power present, which is x³.
In OCR exams, the instruction 'factorise' always implies 'factorise fully'; a partial factorisation (like leaving a factor of 2 inside the bracket) will only score a method mark.
When factorising a quadratic where the constant term (c) is negative, look at the signs immediately: one bracket must contain a '+' and the other a '-'.
Always perform a quick mental expansion of your final brackets; if you don't arrive back at the original expression, you likely have a sign error in your factor pairs.
When factorising by grouping, watch out for negative signs in the second pair of terms. If your two common brackets don't match perfectly, you probably missed a negative common factor.
Factorise
To write an expression as a product of its factors by taking out common terms.
Distributive law
The rule stating that multiplying a number by a group of numbers added together is the same as doing each multiplication separately, used in reverse for factorisation.
Bracket
A pair of symbols used to enclose algebraic terms that are multiplied by a common factor.
Highest Common Factor (HCF)
The largest factor that divides exactly into two or more numerical or algebraic terms.
Term
A single number, a variable, or numbers and variables multiplied together in an algebraic expression.
Coefficient
The numerical multiplier of a variable within an algebraic term.
Monic quadratic
A quadratic expression where the leading coefficient (the number in front of the squared variable) is exactly 1.
Trinomial
An algebraic expression consisting of three distinct terms.
Double brackets
The format of a fully factorised quadratic expression, written as the product of two linear expressions.
Factor pairs
Two numbers that, when multiplied together, result in a specific target product.
Non-monic quadratic
A quadratic expression where the leading coefficient is a number other than 1 or 0.
ac method
A technique used to factorise non-monic quadratics by finding two numbers that multiply to the product of 'a' and 'c' and sum to 'b'.
Splitting the middle term
Rewriting the single 'bx' term of a quadratic as two separate terms to allow for factorisation by grouping.
Factorising by grouping
Splitting a four-term expression into two pairs and taking a common factor from each pair to reveal an identical common bracket.
Put your knowledge into practice — try past paper questions for Mathematics
Factorise
To write an expression as a product of its factors by taking out common terms.
Distributive law
The rule stating that multiplying a number by a group of numbers added together is the same as doing each multiplication separately, used in reverse for factorisation.
Bracket
A pair of symbols used to enclose algebraic terms that are multiplied by a common factor.
Highest Common Factor (HCF)
The largest factor that divides exactly into two or more numerical or algebraic terms.
Term
A single number, a variable, or numbers and variables multiplied together in an algebraic expression.
Coefficient
The numerical multiplier of a variable within an algebraic term.
Monic quadratic
A quadratic expression where the leading coefficient (the number in front of the squared variable) is exactly 1.
Trinomial
An algebraic expression consisting of three distinct terms.
Double brackets
The format of a fully factorised quadratic expression, written as the product of two linear expressions.
Factor pairs
Two numbers that, when multiplied together, result in a specific target product.
Non-monic quadratic
A quadratic expression where the leading coefficient is a number other than 1 or 0.
ac method
A technique used to factorise non-monic quadratics by finding two numbers that multiply to the product of 'a' and 'c' and sum to 'b'.
Splitting the middle term
Rewriting the single 'bx' term of a quadratic as two separate terms to allow for factorisation by grouping.
Factorising by grouping
Splitting a four-term expression into two pairs and taking a common factor from each pair to reveal an identical common bracket.