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)
Evaluate each of these answers:
For example, evaluate
A.
(-1/2)^4 + 7*(1/2)^2 - 9*(1/2) -1
and
0^4 + 7*0^2 - 9*0 -1
If one of these answers is negative and one of these is positive, then the Intermediate Value Theorem guarantees that the polynomial has a root in this interval. If both of the evaluations at the ends of the interval are positive, or both are negative, then the Intermediate Value Theorem does not guarantee that the polynomial has a root in this interval.
Repeat this for B,C, and D, or until you find the right answer.
To determine the interval on which the Intermediate Value Theorem guarantees that the polynomial x^4 + 7x^2 − 9x − 1 has a root, we need to first find the sign changes in the polynomial.
The Intermediate Value Theorem states that if a continuous function changes sign over an interval, then it must have at least one root within that interval.
To find the sign changes, we can observe the signs of the polynomial at the endpoints of each interval.
Let's evaluate the polynomial at the endpoints of each interval:
A. Interval (-1/2, 0):
- Evaluate at x = -1/2:
(-1/2)^4 + 7(-1/2)^2 − 9(-1/2) − 1 = 3/16 > 0
- Evaluate at x = 0:
0^4 + 7(0)^2 − 9(0) − 1 = -1 < 0
There is a sign change between -1/2 and 0. So, the polynomial has at least one root in the interval (-1/2, 0).
B. Interval (1/2, 1):
- Evaluate at x = 1/2:
(1/2)^4 + 7(1/2)^2 − 9(1/2) − 1 = 5/16 > 0
- Evaluate at x = 1:
1^4 + 7(1)^2 − 9(1) − 1 = -2 < 0
There is a sign change between 1/2 and 1. So, the polynomial has at least one root in the interval (1/2, 1).
C. Interval (0, 1/2):
- Evaluate at x = 0:
0^4 + 7(0)^2 − 9(0) − 1 = -1 < 0
- Evaluate at x = 1/2:
(1/2)^4 + 7(1/2)^2 − 9(1/2) − 1 = 5/16 > 0
There is a sign change between 0 and 1/2. So, the polynomial has at least one root in the interval (0, 1/2).
D. Interval (-1, -1/2):
- Evaluate at x = -1:
(-1)^4 + 7(-1)^2 − 9(-1) − 1 = -2 < 0
- Evaluate at x = -1/2:
(-1/2)^4 + 7(-1/2)^2 − 9(-1/2) − 1 = 3/16 > 0
There is a sign change between -1 and -1/2. So, the polynomial has at least one root in the interval (-1, -1/2).
Based on the evaluations, each interval contains a sign change in the polynomial. Therefore, the Intermediate Value Theorem guarantees that the polynomial x^4 + 7x^2 − 9x − 1 has a root in each of these intervals:
A. (-1/2, 0)
B. (1/2, 1)
C. (0, 1/2)
D. (-1, -1/2)
So, the correct answer is all of the above.