Factorising out terms means writing an expression as a multiplication by taking out a common factor shared by every term. It is the reverse of expansion, because instead of distributing a factor into brackets, you collect a shared factor outside the brackets.
A factorised form is useful because multiplication often shows the structure of an expression more clearly than addition or subtraction. In algebra, this can make later work easier, such as simplifying, solving equations, or spotting further factorisation opportunities.
A common factor is something that divides every term in the expression exactly. This common factor may include a number, a variable, or several variables multiplied together.
The highest common factor, often abbreviated to HCF, is the largest factor common to all terms. Taking out the HCF gives the fullest factorisation at this stage, which is why exam questions often ask students to factorise fully.
In algebraic expressions, each term is treated as a product of its parts. For example, a term like contains a numerical part and variable parts, so factorising requires checking both the coefficient and the powers of the variables.
A term that does not visibly show a variable can still be handled correctly by remembering that it simply does not contain that variable as a factor. This is important when deciding whether a variable belongs in the common factor.
The reason factorising works is the distributive law. If , then reversing this gives , so a shared factor can be pulled outside brackets.
This idea applies just as well to subtraction, because . That is why expressions with mixed signs can still be factorised by taking out a common factor.
For variables, the common factor is determined by the lowest power shared by every term. For example, if one term contains and another contains , then the common variable factor in is because that is the largest power dividing both.
The same principle applies independently to each variable. If two terms contain both and , you find the shared power of and the shared power of , then multiply them together with any shared numerical factor.
A useful way to think about factorising is that each term is being rewritten as the common factor multiplied by a leftover part. Those leftover parts become the contents of the bracket, and they must combine correctly when expanded.
Because of this, the bracket terms are not chosen randomly: they are what remain after each original term is divided by the common factor.
Step 1: Separate each term into its parts by identifying the numerical coefficient and the variable factors. This makes it easier to see what is shared across all terms, especially when powers or multiple variables are involved.
Step 2: Find the numerical HCF of the coefficients. This is the largest number that divides every coefficient exactly, and it forms the numerical part of the factor taken outside the brackets.
Step 3: Find the algebraic HCF by checking each variable in turn. A variable can only be included if it appears in every term, and the power used is the smallest power appearing across those terms.
Step 4: Multiply the numerical and algebraic parts to get the full common factor. This gives the largest single factor you can safely take outside the brackets.
Step 5: Divide each original term by the common factor to find the bracket contents. This division must be exact, and it ensures the bracket contains the leftover parts that rebuild the original expression when expanded.
Step 6: Write the expression as . If the question asks to factorise fully, check whether the bracket itself still has a common factor.
Taking out a common factor is often the first stage of factorising, but it is not always the final stage. An expression is fully factorised only when no further common factor or simpler factorisation can be taken from any bracket.
This distinction matters because an answer can be correct in structure but incomplete in form. In exams, incomplete factorisation can lose marks even when the first step is right.
Numerical common factor and algebraic common factor must both be checked. Sometimes students focus only on the coefficients and miss a shared variable, or focus only on variables and miss a larger numerical factor.
The correct HCF is the product of all shared parts, not just one of them. This is why expressions with several variables need a systematic check rather than a quick glance.
The table below shows the main comparisons students need to make when choosing what to factor out.
| Feature | Numerical factor | Variable factor | Full common factor |
|---|---|---|---|
| What to inspect | Coefficients | Variables and powers | Both together |
| Rule used | Greatest common divisor | Lowest shared power | Multiply shared parts |
| Example pattern | |||
| Common mistake | Taking too small a number | Taking the highest power instead of the lowest | Omitting one shared part |
Always look for a common factor before trying any more advanced method. Many algebra questions become much simpler once the greatest common factor is removed, and sometimes this is the entire task.
A quick scan of coefficients and shared variables at the start can save time and prevent unnecessary work.
Factorise fully means keep going until no further common factor remains. After taking out an initial factor, inspect the bracket again in case all bracket terms still share something.
This check is especially important when the first factor taken out was not the highest possible common factor.
Verify by expansion. If you expand your factorised answer and recover the original expression exactly, then the factorisation is algebraically correct.
Expansion is one of the most reliable ways to catch sign mistakes, missing variables, or incorrect bracket terms, so it should be used as a final check whenever possible.
Watch powers carefully. The common factor for variables is based on the smallest shared power, not the largest visible power in the expression.
This is a common exam trap because a large power looks tempting, but it cannot be taken out unless every term contains it.
Using a variable that is not present in every term is a frequent error. If one term lacks a variable entirely, that variable cannot be part of the common factor because dividing that term by the variable would not give an algebraic term of the same type.
This mistake usually comes from noticing a variable repeatedly without checking every term one by one.
Taking the highest power instead of the lowest shared power leads to an invalid factorisation. For example, if the terms contain and , then taking out would not divide the second term exactly.
The common factor must divide all terms fully, so the smallest shared power is the correct choice.
Sign mistakes inside brackets often happen when students mentally remove a factor instead of actually dividing. A negative term inside or outside the bracket affects the remaining signs, so each term should be checked carefully.
When in doubt, divide each term explicitly by the common factor and then expand at the end to confirm the result.
Stopping too early is another misconception. A partially factorised answer may still contain a common factor inside the bracket, meaning the expression is not fully factorised yet.
Full factorisation is reached only when no further common factor or obvious factorisation step remains.
Factorising out terms is the foundation for many later algebra techniques. It appears in simplifying expressions, solving equations by setting factors equal to zero, and preparing expressions for methods such as grouping or quadratic factorisation.
Because of this, mastering common factors is not an isolated skill but a basic tool used across algebra.
The same idea extends beyond two terms and beyond one variable. Whether an expression has two terms or many, the logic is identical: identify what every term shares, remove it, and write the leftovers in brackets.
This makes the method highly transferable, especially in expressions involving several letters and powers.
Factorising is also closely linked to arithmetic structure. Just as a number can be written as a product of its factors, an algebraic expression can often be rewritten to reveal its building blocks.
Seeing algebraic expressions as structured products helps students move from procedural manipulation to deeper mathematical understanding.
