Use the theorem on bounds to establish the best integral bounds for the roots of the equation
w^(4)-8w^(3)+2w^(2)+10w-1=0
Good
To find the appropriate bounds for the roots of the equation you mentioned, we can use a theorem called the Intermediate Value Theorem. This theorem states that if a continuous function has different signs at two points, then it must have at least one root in the interval between these two points.
We start by examining the equation:
w^4 - 8w^3 + 2w^2 + 10w - 1 = 0
The leading coefficient is 1, which means the polynomial is monic and has positive leading terms. Based on the Intermediate Value Theorem, we can conclude that the polynomial has at least one positive root.
Now, let's consider a positive value of w, say w = 1. Plugging it into the equation, we get:
1^4 - 8(1)^3 + 2(1)^2 + 10(1) - 1 = -4
Since the value of the equation at w = 1 is negative, we can conclude that there must be at least one positive root between 0 and 1.
Next, we examine the equation at a negative value of w, say w = -1:
(-1)^4 - 8(-1)^3 + 2(-1)^2 + 10(-1) - 1 = 20
Since the value of the equation at w = -1 is positive, we can conclude that there must be at least one negative root between -1 and 0.
Hence, based on the Intermediate Value Theorem, we can establish the best integral bounds for the roots of the equation:
-1 < w < 0 (negative root)
0 < w < 1 (positive root)