Question

How do you factor `x^3-3x^2+x-55`?

Answer 1

`(x-5)(x^2+2x+11)`

Explanation:

If `x^3-3x^2+x-55` has an integer root `r` then
`r` must be a factor of `55`
i.e. `r in {+-5,+-11}`

A quick check reveals `r=5` satisfies the requirement
so
`(x-r) = (x-5)` is a factor

Dividing `x^3-3x^2+x-55` by `(x-5)`
`color(white)("XXX")`(for example by using synthetic division:)

`{:(," | ", 1, -3,color(white)("X")1, -55), (+5," | ",,color(white)("X")5,10,color(white)("X")55),(,"---","---","---","---","----"),(,,color(red)(1),color(red)(2),color(red)(11),color(white)("X")color(green)(0)):}`

gives
`(x-5)(color(red)(1)x^2+color(red)(2)x+color(red)(11))`

Checking the discriminant of `(x^2+2x+11)` reveals that no further Real factoring is possible.