# How do you use the rational root theorem to find the roots of #x^3 – x^2 – x – 3 = 0#?

##### 2 Answers

The rational root theorem states that any rational root of a polynomial will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In our case

But if you check none of the values above are roots for

So the rational root theorem cannot help us here.

#### Answer:

You can't. You can only use the rational root theorem to show that it has no rational roots.

#### Explanation:

By the rational root theorem, any rational roots of

That means that the only possible rational factors are:

#+-1# ,#+-3#

Let

We find:

#f(1) = 1-1-1-3 = -4#

#f(-1) = -1-1+1-3 = -4#

#f(3) = 27-9-3-3 = 12#

#f(-3) = -27-9+3-3 = -36#

So

In fact it has one Real root:

#x = 1/3 (1+(46-6 sqrt(57))^(1/3)+(46+6 sqrt(57))^(1/3))#

and two Complex roots:

#x = 1/3 (1+omega(46-6 sqrt(57))^(1/3)+omega^2(46+6 sqrt(57))^(1/3))#

#x = 1/3 (1+omega^2(46-6 sqrt(57))^(1/3)+omega(46+6 sqrt(57))^(1/3))#

where

These can be found using Cardano's method or similar.