Final answer:
The function 4(x + 2)(x − 3)^3 is positive when x < -2 or x > 3, determined by analyzing the sign of the function in the intervals created by the zeros of the function.
Step-by-step explanation:
To determine where the function 4(x + 2)(x − 3)^3 > 0, we need to consider the zeros of the function and analyze the sign of the expression in each interval created by these zeros. The zeros of the function occur at x = -2 and x = 3. Since the term (x - 3)^3 is to the third power, the sign of this term will not change when passing through x = 3. However, the term (x + 2) will change its sign when passing through x = -2.
Now, let's assess the sign of the function in the intervals divided by the zeros:
- For x < -2, both factors are negative, resulting in the function being positive.
- For -2 < x < 3, (x + 2) is positive and (x − 3)^3 is negative, thus the function is negative.
- For x > 3, both factors are positive, so the function is again positive.
Therefore, the function is positive when x < -2 or x > 3.