This polynomial is .
Notice that . So is a factor.
Long divide to find the quadratic factor and factorise the quadratic if possible.
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!