What is an example of a quintic polynomial function that has exactly four zeros?
To find an example of a quintic polynomial function with exactly four zeros, we need to construct a polynomial equation of degree 5 that has four distinct solutions.
Let's start by considering the simplest possible scenario: a quintic polynomial with the equation f(x) = (x - a)(x - b)(x - c)(x - d)(x - e), where a, b, c, d, and e are the zeros of the polynomial. We need to make sure that a, b, c, and d are distinct zeros, while e can be repeated.
To illustrate this concept, let's choose the zeros as follows: a = 0, b = 1, c = 2, and d = 3. This gives us a polynomial equation: f(x) = x(x - 1)(x - 2)(x - 3)(x - e), where e can take any value.
For simplicity, let's assume e = 0, so we have: f(x) = x(x - 1)(x - 2)(x - 3)(x - 0).
Expanding this equation, we get: f(x) = x^5 - 6x^4 + 11x^3 - 6x^2.
Therefore, an example of a quintic polynomial function with exactly four zeros (0, 1, 2, and 3) is f(x) = x^5 - 6x^4 + 11x^3 - 6x^2.
To verify this, you can substitute the values of the distinct zeros into the equation and see if they result in f(x) = 0.