SOLUTION: How many roots do the following equations have? -12x^2 -25x+5+x^3=0

1. In the complex plane, this polynomial has three roots, exactly as its degree is.


2. In the real domain, it definitely has at least one root.
   You may conclude it, because the degree of the polynomial is 3 and it tends to  as x --->  and tends to  as x ---> .


3. Now let us check the sign of the polynomial at selected points:

   f(-1) = -12*(-1)^2 - 25*(-1) + 5 - (-1)^3 = 12 + 25 + 5 + 1 = 43 > 0.

   f(1) = -12*1^2 - 25*1 + 5 - 1^3 = -12 - 25 + 5 - 1 = -33 <0.

   It is just enough to conclude that f(x) has three real roots:

      one between  and (-1),     <--- the polynomial change the sign at the ends of this interval;

      second between (-1) and 1,      <---  again, the polynomial change the sign at the ends of this interval;

      third between 1 and .        <--- by the same reason.