I can't see how to factorise this. And can't think of another method. Can someone please help?
x^3 - 12x + 16 = 0
Archie Meade's method assumes that the roots will be integer since those are the only factors he is trying.
Another method is to use the "rational root theorem"- that if m/n is a rational number root of a polynomial with integer coefficients then the denominator, n, must divide the leading coefficient and the numerator, m, must divide the constant term. Here, the leading coefficent is 1 so the denominator must be 1 or -1 (any rational root must be an integer just as Archie Meade thought) and the constant term is 16 so any rational root must evenly divide 16: 1, -1, 2, -2, 4, -4, 8, -8, 16,or -16 are possible roots. Trying each of those numbers in the polynomial we see that and that
Using x-2 and x+ 4 as factors, we see that the third factor is x- 2 so that x= 2 is a double root.
Of course, the rational root theorem can only point out possible rational roots. It may be that a polynomial has no rational number roots but in that case, no method is going to be easy!