Question

How do you solve `x^3-3x^2-9x+27<0` using a sign chart?

Answer 1

The answer is `x in ] -oo,-3 [`

Explanation:

Let `f(x)=x^3-3x^2-9x+27` and `f(x)<0`

Then, `f(-3)=-27-27+27+27=0`

So, `(x+3)` is a factor of `f(x)`

Let's do a long division

`color(white)(aaaa)` `x^3-3x^2-9x+27` `color(white)(aaaa)` `∣` `x+3`
`color(white)(aaaa)` `x^3+3x^2` `color(white)(aaaaaaaaaaaaa)` `∣` `x^2-6x+9`
`color(white)(aaaaa)` `0-6x^2-9x`
`color(white)(aaaaaaa)` `-6x^2-18x`
`color(white)(aaaaaaaaaaa)` `0+9x+27`
`color(white)(aaaaaaaaaaaaa)` `+9x+27`

`x^2-6x+9=(x-3)^2`

So, `f(x)=(x+3)(x-3)^2`

let's do the sign chart

`color(white)(aaaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaaa)` `-3` `color(white)(aaaaa)` `3` `color(white)(aaaaa)` `+oo`
`color(white)(aaaaa)` `x+3` `color(white)(aaaaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`
`color(white)(aaaaa)` `(x-3)^2` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`
`color(white)(aaaaa)` `f(x)` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

`f(x)<0` when `x in ] -oo,-3 [`

graph{x^3-3x^2-9x+27 [-76.4, 55.3, -32.1, 33.74]}