Final answer:
The function f(x) = -3x(x^2 - 9)(x^2 + 1) has roots where each factor equals zero. It has real roots at x = -3, x = 0, and x = 3, while x^2 + 1 does not contribute any real roots since it's always positive.
Step-by-step explanation:
The function given is f(x) = -3x(x2 - 9)(x2 + 1). To fully understand this function, let's look at each factor individually. The function can be analyzed by identifying its factors and their roots.
The first factor, -3x, has a root at x = 0. The second factor, (x2 - 9), factors further into (x + 3)(x - 3), with roots at x = -3 and x = 3. The third factor, (x2 + 1), has no real roots because x2 + 1 is never zero for any real number x.
The roots of the function are the values of x for which the function equals zero. Therefore, the roots of the function are x = -3, x = 0, and x = 3. No other values of x will make the function equal to zero, as x2 + 1 remains positive for all real x.