First, we should always check for any common factors. In this example, there aren't any.
So is this a difference of two squares? Clearly not, because there are three terms. It's a trinomial.
The coefficient of is 1 and the other coefficients are positive, so this is the easiest type of trinomial.
We just need to find two numbers that multiply to 10 and add to 11. The factor pairs of 10 are 25 (but those add to 7) and 110 (which is the pair we want, as they add to 11).
So
Example 6 (non-calculator)
Factorise
This is a trinomial expression with no common factors.
The coefficient of is 1, so this is a unitary trinomial (also called a monic trinomial).
So we just need to find two numbers that multiply to –8 and add to –2. Clearly one of them will be negative.
The only two numbers that fit these conditions are 2 and –4.
A question of this type has never appeared in a National 5 Maths exam, but it might, so let's be ready for it!
Rearranging the terms:
The coefficient of is –1 so one way of handling this is to take out a common factor of –1.
We can then factorise the unitary trinomial in the brackets in the usual way, looking for two numbers that multiply to –12 and add to –4.
The two numbers are –6 and 2.
Putting this all together:
We think it is fine to leave this with the negative sign at the beginning, but if you don't like that style, you could multiply the factor through by –1 and present your answer in the tidier form:
First check for any common factors. There aren't any.
The coefficient of is 2. That's good news, because 2 is a prime number. The only possible product is 12.
The constant term of –4 isn't such good news. 4 isn't prime. It could be 14 or 22.
22 wouldn't work here because the outer pair and inner pair would then be an even number, not –7. So it must be 14. Checking this both ways around and taking care over the signs gives us:
So
There are several ways to make this more systematic, and different teachers prefer different methods. We like what is called the "ac method". More on that in the resources below.
Example 12 (non-calculator)
Fully factorise
First check for any common factors. All of the coefficients are multiples of 3, so we factor that out.
So
The trinomial in the brackets is the same as the previous example, so the full factorisation is:
This is very similar to the previous example. The answer has two steps. First we remove the common factor and then we factorise the difference of two squares.
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