Factorising Algebra
Factorisation doesn't just work with numbers; it can be used with algebra
as well. With this example we find common factors (one step at a time). First the coefficient, then the variable:
48x^{2} –
3x^{6
}= 3(16x^{2} – x^{6})
= 3x^{2}(16 – x^{4}) 
So the original expression 12x^{2} – 3x^{6} has factors of 3x^{2} and 16 – x^{4}. Now, 16 – x^{4} is in a pattern that we should recognise (the difference of two squares), and can also be factorised. The two factors will be very similar, but with a difference of sign for the x terms. It's easier to see this going in reverse – a process called expanding –
with an extra line of working shown in between, which we'll include here. Note that the x^{2} terms cancel each other out.
16
 x^{4
} = 16  4x^{2} + 4x^{2} – x^{4}
^{ }= (4 + x^{2})(4 – x^{2}) 
Notice that 4 – x^{2} is in the same pattern,
so we have another factor which can itself be factorised.
4
– x^{2}
= 4 – 2x + 2x – x^{2}
= (2 + x)(2 – x) 
And so we finally have our answer. It doesn't matter which order the
factors are written in (because multiplication is commutative). The 3x^{2} could
be put within its own brackets, but it normally is not.
48x^{2} –
3x^{6
}= 3(16x^{2} – x^{6})
= 3x^{2}(16 – x^{4})
= 3x^{2}(4 + x^{2})(4 – x^{2})
= 3x^{2}(4 + x^{2})(2 + x)(2
– x) 
So the factors of 48x^{2} – 3x^{6} are 3x^{2}, 4
+ x^{2}, 2 + x, and 2 – x.
