How do you use the rational roots theorem to find all possible zeros of $x^{3} - x^{2} - 4 x + 4$ $x^3-x^2-4x+4$ ?

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Answer 1 · 2016-08-20T21:35:52

$x \in {- 2, + 1, + 2}$ $color(green)(x in {-2, +1, +2})$
(see below for use of the Rational Root Theorem)

The Rational Root Theorem tells us that the rational zeros of

$1 x^{3} - x^{2} - 4 x + 4$ $color(red)(1)x^3-x^2-4x+color(blue)(4)$
are the factors of $\frac{4}{1}$ $(color(blue)(4))/(color(red)(1))$

Checking each of the possible factors we get:
enter image source here
which gives us zeros for

$x = - 2, x = 1, and x = 2$ $x=-2, x=1, and x=2$

Since $x^{3} - x^{2} - 4 x + 4$ $x^3-x^2-4x+4$ is of degree $3$ $3$
it can have a maximum of $3$ $3$ zeros
and we have found them all.

How do you use the rational roots theorem to find all possible zeros of x3−x2−4x+4x^3-x^2-4x+4?