Find a bound on the real zeros of the polynomial function f(x)=x^4+x^3-4x-6
Every real zero of f lies between ______ and______.
To find a bound on the real zeros of the polynomial function f(x) = x^4 + x^3 - 4x - 6, we can use the Intermediate Value Theorem along with some analysis of the function.
1. First, let's evaluate the function f(x) at different values to determine its behavior. We can start by calculating f(0):
When x = 0, f(x) = (0)^4 + (0)^3 - 4(0) - 6 = -6.
Hence, f(0) = -6.
2. Next, let's evaluate the function f(x) for some other values. We can substitute different values of x into the function to observe whether the function changes sign:
When x = 1, f(x) = (1)^4 + (1)^3 - 4(1) - 6 = -8.
Hence, f(1) = -8.
When x = 2, f(x) = (2)^4 + (2)^3 - 4(2) - 6 = 14.
Hence, f(2) = 14.
3. By evaluating the function at different x-values, we see that f(x) changes sign between x = 0 and x = 1, indicating that there is at least one real zero in that interval.
4. To further narrow down the bounds, we can look for additional sign changes in the function. Let's evaluate f(x) at x = -1:
When x = -1, f(x) = (-1)^4 + (-1)^3 - 4(-1) - 6 = -2.
Hence, f(-1) = -2.
5. By observing the change in sign between f(0) = -6 and f(-1) = -2, we can deduce that there is another real zero between x = -1 and x = 0.
6. Therefore, a bound on the real zeros of f(x) = x^4 + x^3 - 4x - 6 is:
-1 < x < 0.