If alpha, beta, gamma and Delta are the roots of the equation x^4+px^3+qx^2+rx+s=0 then (summation of[ (alpha, beta) /r^2])
does "(summation of[ (alpha, beta) /r^2])" mean (alpha+beta)/r^2 ??
and if so, I suspect it depends on which two roots you pick.
We do know that the sum of all the roots is -p
To find the value of the expression, we need to know the values of alpha, beta, gamma, and Delta. However, the given equation only provides their relationship with the coefficients p, q, r, and s.
To proceed, we'll need to use Vieta's formulas, which relate the coefficients of a polynomial equation to its roots.
Vieta's formulas state that for a polynomial equation of the form:
ax^n + bx^(n-1) + cx^(n-2) + ... + k = 0
The sum of the roots is equal to the negation of the coefficient of the second-to-last term, divided by the leading coefficient (a):
Sum of roots = -b/a
The product of the roots taken two at a time, denoted as sigma(ab), is equal to the negation of the coefficient of the third-to-last term, divided by the leading coefficient (a):
sigma(ab) = c/a
Now let's apply Vieta's formulas to the given equation:
x^4 + px^3 + qx^2 + rx + s = 0
According to Vieta's formulas, we can determine the following relationships:
alpha + beta + gamma + Delta = -p (sum of roots)
alpha * beta + alpha * gamma + alpha * Delta + beta * gamma + beta * Delta + gamma * Delta = q (sum of pairwise products)
alpha * beta * gamma + alpha * beta * Delta + alpha * gamma * Delta + beta * gamma * Delta = -r (sum of triple products)
alpha * beta * gamma * Delta = s (product of roots)
However, the expression you provided is the summation of (alpha, beta) / r^2. It seems like there might be a mistake or omission in the expression.
Could you please provide the correct expression, or clarify any missing information?