Final answer:
The equation of the degree 5 polynomial function with a double zero at x = 1, a triple zero at x = 3, and passes through the point (2, 53) is (x - 1)²(x - 3)³ * 53.
Step-by-step explanation:
The equation of a degree 5 polynomial function with a double zero at x = 1 and a triple zero at x = 3 can be written as (x - 1)²(x - 3)³g(x), where g(x) is a polynomial of degree 5 - 2 - 3 = 0. Since the function passes through the point (2, 53), we can substitute x = 2 and y = 53 into the equation to solve for the constant term, g(2). Plugging in the values gives us (2 - 1)²(2 - 3)³g(2) = 0 * g(2) = 53. This implies that g(2) = 53.
Therefore, the equation of the polynomial function is f(x) = (x - 1)²(x - 3)³ * 53.