On which interval does the Intermediate Value Theorem guarantee that the polynomial x^4 + 7x^2 − 9x −
1 has a root?
A. (-1/2,0)
B. (1/2,1)
C. (0,1/2)
D. (-1,-1/2)
To find the interval on which the Intermediate Value Theorem guarantees that the polynomial has a root, we need to evaluate the polynomial at the endpoints of each given interval and check if the signs of the values are different.
Let's evaluate the polynomial at the endpoints of each interval:
A. (-1/2, 0):
- Plug -1/2 into the polynomial: (-1/2)^4 + 7(-1/2)^2 − 9(-1/2) − 1 = 1/16 + 7/4 + 9/2 - 1 = 1/16 + 28/16 + 18/16 - 16/16 = 31/16
- Plug 0 into the polynomial: 0^4 + 7(0)^2 − 9(0) − 1 = 0 - 1 = -1
Since the signs of the values at the endpoints (-1 and 0) are different (positive and negative), the Intermediate Value Theorem guarantees that the polynomial has a root in the interval (-1/2, 0).
B. (1/2, 1):
- Plug 1/2 into the polynomial: (1/2)^4 + 7(1/2)^2 − 9(1/2) − 1 = 1/16 + 7/4 - 9/2 - 1 = 1/16 + 28/16 - 18/16 - 16/16 = -5/16
- Plug 1 into the polynomial: 1^4 + 7(1)^2 − 9(1) − 1 = 1 + 7 - 9 - 1 = -2
Since the signs of the values at the endpoints (1/2 and 1) are the same (negative), the Intermediate Value Theorem does not guarantee that the polynomial has a root in the interval (1/2, 1).
C. (0, 1/2):
- Plug 0 into the polynomial: 0^4 + 7(0)^2 − 9(0) − 1 = 0 - 1 = -1
- Plug 1/2 into the polynomial: (1/2)^4 + 7(1/2)^2 − 9(1/2) − 1 = 1/16 + 7/4 - 9/2 - 1 = 1/16 + 28/16 - 18/16 - 16/16 = -5/16
Since the signs of the values at the endpoints (0 and 1/2) are the same (negative), the Intermediate Value Theorem does not guarantee that the polynomial has a root in the interval (0, 1/2).
D. (-1, -1/2):
- Plug -1 into the polynomial: (-1)^4 + 7(-1)^2 − 9(-1) − 1 = 1 + 7 + 9 - 1 = 16
- Plug -1/2 into the polynomial: (-1/2)^4 + 7(-1/2)^2 − 9(-1/2) − 1 = 1/16 + 7/4 + 9/2 - 1 = 1/16 + 28/16 + 18/16 - 16/16 = 31/16
Since the signs of the values at the endpoints (-1 and -1/2) are the same (positive), the Intermediate Value Theorem does not guarantee that the polynomial has a root in the interval (-1, -1/2).
Based on the evaluation of each interval, the Intermediate Value Theorem guarantees that the polynomial x^4 + 7x^2 − 9x − 1 has a root in the interval (-1/2, 0). Therefore, the correct answer is A. (-1/2, 0).