# How do you use the rational root theorem to find the roots of x^3-x^2+2x-2?

May 22, 2015

Answer: The root of this polynomial is $1$.

The Rational Roots Theorem says that:

• if $P \left(x\right)$ is a polynomial with integer coefficients
• and $\frac{p}{q}$ is a root of $P$ (i.e. $P \left(\frac{p}{q}\right) = 0$ ),

then $p$ is a factor of the constant term of $P$ and $q$ is a factor of the leading coefficient of $P$.

In our case, $P \left(x\right) = {x}^{3} - {x}^{2} + 2 x - 2$. So, the constant term is $- 2$ and the leading coefficient is $1$.

First, let's write down all the factors of the constant term:
$\pm 1 , \mathmr{and} \pm 2$. These will be the possible values of $p$.

Next, let's write down all the factors of the leading coefficient:
$\pm 1$. These will be the possible values of $q$.

Now, let's write down the possible values of $\frac{p}{q}$:
$\pm \frac{1}{1} , \pm \frac{2}{1}$, which can be simplified as $\pm 1 \mathmr{and} \pm 2$.

It can be easily verified that the only $\frac{p}{q}$ that is a rational root of P is $1$.

(We can check this result by factoring P(x) as ${x}^{2} \left(x - 1\right) + 2 \left(x - 1\right) = \left(x - 1\right) \left({x}^{2} + 2\right)$, which admits the solution $1$)

A graphical illustration can be seen below, by plotting the corresponding function:

graph{x^3 - x^2 + 2x - 2 [-10, 10, -5, 5]}