Question

How do you find all zeros for `x^3 - 5x^2 + x - 5`?

Answer 1

`x =5`
or
`x=+-i`

Explanation:

`x^3-5x^2+x-5` can be factored as
`color(white)("XXX")color(red)(""(x^3+x))-color(blue)(""(5x^2+5))`

`color(white)("XXX")=color(red)(x(x^2+1))-color(blue)(5(x^2+1)`

`color(white)("XXX")=(color(red)(x)-color(blue)(5))*(x^2+1)`

For the zeroes:
either
`color(white)("XXX")(x-5)=0color(white)("XX")rarrcolor(white)("XX")x=5`
or
`color(white)("XXX")(x^2+1)=0color(white)("XX")rarrcolor(white)("XX")x=+-sqrt(-1)=+-i`