Question

Show that the equation x^3+ x^2-3=0 has no rational roots, but that it does have an irrational root between x=1 and x=2

Answer 1

if x^3 + x^2 - 3 = 0
has any rational roots then they must be one of
-3 or +3
that is there should be a factor of either (x+3) or (x-3)
Using either long algebraic division, or synthetic division, it can be shown that neither divides into it evenly.
However,
f(1) = 1 + 1 - 3 = -1
f(2) = 8 + 4 - 3 = 9
Since the curve it continuous, and at x = 1 it is below the x-axis, and at x = 2 it lies above the x-axis, for some value of x between 1 and 2 it must have crossed the x-axis, which would be the root.

A quick sketch will show that it can cross only once,

http://www.wolframalpha.com/input/?i=x%5E3%2B+x%5E2+-+3+%3D+0