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).
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}.
But 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.
16
 x^{4
} = 16  4x^{2} + 4x^{2}  x^{4}
^{ }= (4 + x^{2})(4  x^{2}) 
But 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) and the 3x^{2} can
be put within its own brackets, but it doesn't have to be.
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
