What is Factorising?
Factorising is the process of writing an algebraic expression as a product of factors – similar to how the number 15 can be written as 5 x 3, where 5 and 3 are both factors of 15. Factorising algebraic expressions involves using one or more brackets (parentheses). The number of brackets required depends on the type of expression being factorised. Factorising is the reverse process of expanding (multiplying out) brackets. So, you can check if factorising has been done correctly by multiplying out your answer to see if it matches the original expression. Factorising is sometimes called factoring.
The Three Types
Algebraic expressions are factorised into one of three types – common factor, difference of two squares or trinomials. More generally trinomials are an example of a polynomial expression and these have their own techniques for factorising such as synthetic division or algebraic long division – but these are beyond the scope of what we’re looking at here. The first key to mastering factorising is knowing that there are three types and how to recognise them. Each of the three types of expression has features that will help you identify it. However, every algebraic expression looks a little different so it takes some experience to accurately identify the types.
Trinomials
Trinomials are probably the most easily identified. ‘Tri’ means three and ‘nomial’ refers to the power (exponent). So a ‘trinomial’ refers to an expression with three terms, each with different powers. A trinomial is a type of quadratic expression since the highest power (called the order) is 2. A trinomial has one term with a power of 2, one term with a power of 1, and one term with a power of 0 (which is the number term on the end). The following are all examples of trinomials. Although each looks a little different they share the same features which defines them as a trinomial.
A Difference of Two Squares
A difference of two squares literally means ‘something squared minus something squared’ since ‘difference’ means substract in Math. So this type of expression has two squared terms with a minus between them. Often, differences of two squares have one algebraic term and one number term. The number term generally won’t have the power 2 on it – instead the number will actually be squared. For example instead of writing 4^2 (i.e. 4 squared) it would be wrtten as the result of 4 squared which is 16. Because a difference of two squares contains a squared term it is also an example of a quadratic expression like trinomials. The following are all examples of differences of two squares, sharing the same common features, but each is just a little different.
A Common Factor
Algebraic expressions which contain common factors are less easily recognisable than trinomials or differences of two squares. This is because they can appear in a range of different ways. In fact, it’s possible for both trinomials and differences of two squares to also contain common factors, and they often do. Sometimes it’s best to identify a common factor type factorising by process of elimination i.e. if it’s not a trinomial or difference of two squares it must be a common factor as that’s the only other option if it can be factorised. An important point with this type is that sometimes the common factor is a number, sometimes a letter, and sometimes both. The following are all examples of expressions which can be factorised using a common factor. Note the wide range of formats.
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Factorising Trinomials
Of the three types trinomials are generally considered the most difficult to factorise. But the difficulty depends on the coefficient numbers in the expression and whether they are positive or negative. For example, small positive numbers are relatively straight-forward but a mix of large and small, positive and negative, is much more difficult to factorise. Check out our YouTube Video on Factorising for a complete explanation of factorising trinomials.
Factorising A Difference of Two Squares
In many cases a difference of two squares is quite straight-forward to factorise since the expression is similar each time as is the format of the factors. A difference of two squares factorises into two brackets with the first term in each bracket being the same and the second terms also being the same. However one bracket contains a ‘plus’ and the other a ‘minus’. This format never changes for a difference of two squares. Check out our YouTube Video on Factorising for a complete explanation of factorising a difference of two squares.
Factorising Using a Common Factor
Anytime you need to factorise you should always check for a common factor – even if the expression is also a trinomial or difference of two squares. Removing the common factor first makes factorising these expressions much easier (although they can be factorised without removing the common factor). An expression can sometimes contain more than one common factor, but can only ever have one ‘highest’ common factor. It’s the highest common factor that we look for and this goes outside (in front) of one bracket. Check out our YouTube Video on Factorising for a complete explanation of factorising using a common factor.
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