Final answer:
The end behavior of the polynomial p(x) = 3x³ - 500x² is that as x approaches negative infinity, the value of p(x) approaches negative infinity, and as x approaches positive infinity, the value of p(x) approaches positive infinity.
Step-by-step explanation:
The end behavior of a polynomial is determined by the highest power term in the polynomial and its coefficient. In the given polynomial p(x) = 3x³ - 500x², the highest power term is 3x³.
As the exponent of the highest power term is odd, the end behavior of the polynomial is opposite for large negative and large positive values of x. This means that as x approaches negative infinity, the value of p(x) approaches negative infinity, and as x approaches positive infinity, the value of p(x) approaches positive infinity.
Graphically, this means that the curve of the polynomial will approach the x-axis as x goes to negative infinity and as x goes to positive infinity. The coefficient of the highest power term, 3, determines how fast the curve approaches or diverges from the x-axis as x becomes large.