Question

How do you factor `x^3-12x^2+41x-42`?

Answer 1

`(x-2)(x-3)(x-7)`

Explanation:

If the roots of the cubic are integer, then they must divide into 42. Possible candidates, plus or minus: 1, 2, 3, 6, 7, 14, 21, 42.

A little exploration with a calculator reveals that both `x=2` and `x=3` are roots, so we can immediately factor the expression as
`(x-2)(x-3)(x-7)`, deducing from `42=2*3*7` that the third root must be 7.