Final answer:
To find where function r, r(x) = (x³-4x)/(3x⁴-2x-4), equals zero, we solve x³-4x = 0. The solutions are x = 0, x = 2, and x = -2, which are the values of x that make r(x) equal to zero.
Step-by-step explanation:
To find the values of x for which the function r, given by r(x) = (x³-4x)/(3x⁴-2x-4), equals zero, we set the numerator equal to zero because a fraction equals zero if and only if its numerator is zero. Thus, we solve the equation x³-4x = 0.
Factoring out an x, we get x(x²-4) = 0. This further factors to x(x-2)(x+2) = 0. Setting each factor equal to zero gives the solutions x = 0, x = 2, and x = -2.
The values of x for which r(x) = 0 are therefore 0, 2, and -2.