How to use Rolle's string to determine how many roots the polynomial 3x^4-4x^3-12x^2+4=0 has?

First, we'll check the continuity of the function. The
Rolle's string could be applied if and only if the polynomial function is
continuous.

lim f(x) = lim (3x^4-4x^3-12x^2+4) = +
infinite, for x approaches to -infinite or
x->+infinite.

To determine the Rolle's string we
need to determine the roots of the 1st derivative of the
function.

f'(x) = 12x^3 - 12x^2 -
24x

We'll put f'(x) = 0

12x^3
- 12x^2 - 24x = 0

We'll factorize by
12x:

12x(x^2 - x - 2) = 0

12x
= 0 <=> x1 = 0

x^2 - x - 2 =
0

x2 = [1 + sqrt(1 + 8)]/2

x2
= (1+3)/2

x2 = 2

x3 =
-1

Now, we'll calculate the values of the function for each
value of the roots of the derivative.

f(-1) = 3*(-1)^4 -
4*(-1)^3 - 12*(-1)^2 + 4

f(-1) = 3 + 4 - 12 + 4 =
-1

f(0) = 4

f(2) =
-28

The values of the function represents the Rolle's
string.

+inf.  -1  +4  -28 
+inf.

We notice that the sign is changing 4
times:

- from +inf. to -1

-
from -1 to +4

- from +4 to
-28

- from -28 to
+inf.

Therefore, the equation will have 4
real roots, located each in the intervals: (-inf. ; -1) ; (-1 ; 0) ; (0 ; 2) ; (2 ;
+inf.).