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 w^4 - 8w^3 + 2w^2 + 10w - 1 = 0, we can make use of the theorem on bounds, also known as the Rational Root Theorem or the Integer Root Theorem. This theorem helps us find potential rational roots of a polynomial equation.
The Rational Root Theorem states that if a polynomial equation has a rational root p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p should be a divisor of the constant term and q should be a divisor of the leading coefficient.
In our case, the polynomial equation is w^4 - 8w^3 + 2w^2 + 10w - 1 = 0. The leading coefficient is 1, and the constant term is -1 (since the last term is -1).
According to the theorem, the possible rational roots are obtained by taking the factors of the constant term (-1) and dividing them by the factors of the leading coefficient (1). In this case, the factors of -1 are ±1, while the factors of 1 are ±1.
Therefore, the potential rational roots are ±1/1 = ±1.
Now, we can test each of these potential roots (±1) in the original equation to determine if they are solutions of the equation. By substituting w = 1, we find:
(1)^4 - 8(1)^3 + 2(1)^2 + 10(1) - 1 = 1 - 8 + 2 + 10 - 1 = 4. Since the result is not zero, w = 1 is not a root.
By substituting w = -1, we find:
(-1)^4 - 8(-1)^3 + 2(-1)^2 + 10(-1) - 1 = 1 + 8 + 2 - 10 - 1 = 0. We have found a root, w = -1.
Now, we divide the original equation by (w + 1) to obtain the reduced cubic equation: w^3 - 9w^2 + 11w - 1 = 0.
Next, we can repeat the process by finding the potential rational roots of this reduced cubic equation (-1, 1, -1/1, 1/1) and testing them in the same way. However, to simplify the process, we can use another theorem called Descartes' Rule of Signs, which helps determine the number of positive and negative roots of a polynomial equation.
By applying Descartes' Rule of Signs to the reduced cubic equation, we determine that there is only one sign change in the coefficients. This means there is exactly one positive root.
Now, we can test the negative potential root, w = -1. By substituting w = -1 in the reduced cubic equation, we find:
(-1)^3 - 9(-1)^2 + 11(-1) - 1 = -1 - 9 - 11 - 1 = -22. Since the result is not zero, w = -1 is not a root of the reduced cubic equation.
Therefore, after testing the potential rational roots, we have found that the equation w^4 - 8w^3 + 2w^2 + 10w - 1 = 0 does not have any integral roots within the bounds ±1.