Show that the equation x3 x2 2x 1 0 has no positive squa

Show that the equation x^3 - x^2 + 2x + 1 = 0 has no positive squareroots.

Solution

As given hint, we multiply given expression by (x+1) as :

(x+1)(x^3-x^2+2x+1) = x(x^3-x^2+2x+1) +1(x^3-x^2+2x+1)

= x^4-x^3+2x^2+1+x^3-x^2+2x+1

=x^4+x^2+2x+2

that still does not satisfy by any positive real value of x, so clearly given equation is satisfied with only negative roots only.