Final answer:
The intermediate value theorem states that if a function is continuous on a closed interval [a, b], and the function takes on values f(a) and f(b) such that f(a) does not equal f(b), then the function must take on every value in between f(a) and f(b) at least once. In this case, the function values at -4 and -3 have opposite signs, indicating the presence of a real zero between -4 and -3.
Step-by-step explanation:
The intermediate value theorem states that if a function is continuous on a closed interval [a, b], and the function takes on values f(a) and f(b) such that f(a) does not equal f(b), then the function must take on every value in between f(a) and f(b) at least once.
In this case, we can find the values of the function at the given integers -4 and -3:
f(-4) = 3*(-4)⁴ - 9*(-4)³ + 5*(-4) - 8 = 128 - 432 - 20 - 8 = -332
f(-3) = 3*(-3)⁴ - 9*(-3)³ + 5*(-3) - 8 = 81 - 243 - 15 - 8 = -185
Since the function values at -4 and -3 have opposite signs (-332 < 0 and -185 > 0), we can conclude that there is a real zero between -4 and -3 by the intermediate value theorem.