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

1 Answer
May 22, 2015

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

The Rational Roots Theorem says that:

  • if #P(x)# is a polynomial with integer coefficients
  • and #p/q# is a root of #P# (i.e. #P(p/q) = 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(x) = x^3-x^2+2x-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:
#+-1, and +-2#. These will be the possible values of #p#.

Next, let's write down all the factors of the leading coefficient:
#+-1#. These will be the possible values of #q#.

Now, let's write down the possible values of #p/q#:
#+-1/1, +-2/1#, which can be simplified as #+-1 and +-2#.

It can be easily verified that the only #p/q# that is a rational root of P is #1#.

(We can check this result by factoring P(x) as #x^2(x-1) +2(x-1) = (x-1)(x^2+2)#, 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]}