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
To establish the best integral bounds for the roots of the equation, we can use the Rational Root Theorem, which provides a list of possible rational roots to test.
The Rational Root Theorem states that if a polynomial equation has a rational root p/q, where p and q are integers, then p must be a factor of the constant term (in this case, -1) and q must be a factor of the leading coefficient (in this case, 1).
In this equation, the constant term is -1, and the leading coefficient is 1. Therefore, we need to find the factors of -1 and 1, which are:
- Factors of -1: ±1
- Factors of 1: ±1
So, the possible rational roots are ±1.
To determine the best integral bounds for the roots, we can use the Intermediate Value Theorem. This theorem states that for a continuous function, if f(a) and f(b) have opposite signs, then the function must have at least one root between a and b.
Let's evaluate the given polynomial equation at the possible rational roots:
For w = -1:
(-1)^4 - 8(-1)^3 + 2(-1)^2 + 10(-1) - 1 = 1 + 8 + 2 - 10 - 1 = 0. Since the result is 0, -1 is a root of the equation.
For w = 1:
1^4 - 8(1)^3 + 2(1)^2 + 10(1) - 1 = 1 - 8 + 2 + 10 - 1 = 4. Since the result is positive, 1 is not a root of the equation.
Based on these evaluations, we have found that -1 is a root of the equation. To find the remaining roots, we can use polynomial long division or synthetic division to divide the given polynomial by (w + 1). This will yield a cubic equation, which we can solve to find the other roots.
Therefore, the best integral bounds for the roots of the equation w^4 - 8w^3 + 2w^2 + 10w - 1 = 0 are (-∞, -1], (-1, +∞).
To establish the best integral bounds for the roots of the equation w^4 - 8w^3 + 2w^2 + 10w - 1 = 0, we can use the theorem on bounds, also known as the Rational Root Theorem.
According to the Rational Root Theorem, if a polynomial equation has a rational root in the form p/q, where p is a factor of the constant term (-1 in this case) and q is a factor of the leading coefficient (1 in this case), then p must be a factor of the constant term and q must be a factor of the leading coefficient.
In this case, the constant term is -1, which has the factors ±1 and ±1. The leading coefficient is 1, which also has the factors ±1 and ±1. Therefore, the possible rational roots will be:
p/q = ±1, ±1
Now we can test these possible roots using synthetic division or any other suitable method to see which ones actually satisfy the equation.