# 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.

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.