Find the bounds on the real zeros of the following function.
f(x)=x^5-5x^4+8x+2
A little synthetic division will show that all real roots r of f(x) satisfy
-1 < r < 5
To find the bounds on the real zeros of the function f(x) = x^5 - 5x^4 + 8x + 2, we can use the Intermediate Value Theorem and the Descartes' Rule of Signs.
1. Intermediate Value Theorem:
We start by evaluating the function at specific points to determine if it changes sign. To do this, we can plug in some values into the function and check the sign of the result.
a) Plug in a value less than -2, let's say -3:
f(-3) = (-3)^5 - 5(-3)^4 + 8(-3) + 2 = -64 + 405 - 24 + 2 = 319
Since f(-3) is positive (greater than zero), it means the function changes sign somewhere between -3 and 0.
b) Plug in a value between -2 and 0, such as -1:
f(-1) = (-1)^5 - 5(-1)^4 + 8(-1) + 2 = -1 - 5 + 8 + 2 = 4
Since f(-1) is positive, it means the function changes sign between -2 and 0.
c) Plug in a value greater than 0, let's say 1:
f(1) = (1)^5 - 5(1)^4 + 8(1) + 2 = 1 - 5 + 8 + 2 = 6
Since f(1) is positive, it means the function also changes sign somewhere between 0 and 1.
From the intermediate value theorem, we can conclude that the function has at least two real zeros—one between -3 and 0, and another between 0 and 1.
2. Descartes' Rule of Signs:
Next, we can use Descartes' Rule of Signs to determine the maximum possible number of positive and negative real zeros the function can have.
a) Count the number of sign changes:
Looking at the function f(x) = x^5 - 5x^4 + 8x + 2, we observe two sign changes. This means there can be a maximum of two positive real zeros.
b) Evaluate the function at -x:
For f(-x), we replace x with -x:
f(-x) = (-x)^5 - 5(-x)^4 + 8(-x) + 2 = -x^5 - 5x^4 - 8x + 2
The signs are: - + - + +
Again, there are two sign changes. Hence, by Descartes' Rule of Signs, there could be a maximum of two negative real zeros.
3. Conclusion:
Based on the Intermediate Value Theorem and Descartes' Rule of Signs, we can conclude that the function f(x) = x^5 - 5x^4 + 8x + 2 has either one or three real zeros, with possibly one positive real zero and two negative real zeros. The exact values and precise locations of these zeros may require further analysis or numerical approximation methods.