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

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### 1 Answer

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.).**