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.