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 use the theorem on bounds, also known as the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial equation with integer coefficients has a rational root p/q (where p and q are integers with no common factors), 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 our equation, the constant term is -1, and the leading coefficient is 1. Therefore, the possible rational roots of the equation are all the factors of -1 divided by the factors of 1.
The factors of -1 are 1 and -1, and the factors of 1 are also 1 and -1. So, the potential rational roots are 1, -1.
Now, we can test these values by substituting them into the equation to see if they indeed satisfy the equation.
For w = 1:
(1)^(4) - 8(1)^(3) + 2(1)^(2) + 10(1) - 1 = 0
1 - 8 + 2 + 10 - 1 = 0
4 ≠ 0
Therefore, w = 1 is not a root of the equation.
For w = -1:
(-1)^(4) - 8(-1)^(3) + 2(-1)^(2) + 10(-1) - 1 = 0
1 + 8 + 2 - 10 - 1 = 0
0 = 0
Therefore, w = -1 is a root of the equation.
So, we have found one root, w = -1, which means the polynomial equation can be factored as (w + 1)(w^3 + ... ) = 0.
To find the remaining roots, we can divide the polynomial equation by (w + 1) and solve the resulting cubic equation.
In summary, the best integral bounds for the roots of the equation w^(4) - 8w^(3) + 2w^(2) + 10w - 1 = 0 are w = -1. To find the remaining roots, we need to further factor the equation.