Final answer:
The degree of the given polynomial is 23. The most total number of peaks and valleys of the graph of the polynomial is 1. The x-intercepts of k(x) are 0 and ±√4.
Step-by-step explanation:
a. Degree of the polynomial:
The degree of a polynomial is the highest power of the variable in the polynomial. In the given polynomial, k(x) = x²³ - 4x², the highest power of x is 23. Therefore, the degree of the polynomial is 23.
b. Total number of peaks and valleys:
The number of peaks and valleys of a polynomial can be determined by analyzing its graph. The given polynomial has an odd degree (23), which means it will have exactly one peak or valley. So, the most total number of peaks and valleys of the graph of the given polynomial is 1.
c. x-intercepts of k(x):
The x-intercepts of a polynomial can be found by setting the polynomial equal to zero and solving for x. Setting k(x) = 0, we get x²³ - 4x² = 0. Factoring out x², we have x²(x²¹ - 4) = 0. Therefore, the x-intercepts of k(x) are x = 0 and x = ±√4.
d. Graph of the polynomial:
The general shape of the given polynomial can be represented by its degree. Since the degree is odd (23), the graph will rise to the left and right as x approaches negative and positive infinity. However, without further information about the coefficients and specific values of the polynomial, we cannot determine the exact shape of the graph.
Learn more about Polynomial degree and graph analysis