User:Mathy Oxalis/sandbox

Young's Lemma Of Sums Of The Multiplied Combinations Of The Roots Of A Polynamial
This lemma was discovered by Alex Young in early 2016.

What the lemma states:

Example:

Consider the polynomial: P(x) = 2x^4 - 19x^3 + 61x^2 -74x + 24 = 0 Where the roots are:

x = 1/2, 2 , 3 , 4

Where 1/2 = r_1, 2 = r_2 , 3 = r_3 , 4 = r_4

And we want to find the value of:

r_1r_2r_3 + r_1r_2r_4 + r_1r_3r_4 + r_2r_3r_4

We can use the lemma to find this value. We see that this is a 4th degree polynomial, and in each combination there are 3 terms, 4-3=1, so we look at the coefficient of x (x^1), which is -74, we also see that the coefficient of x^4 (The highest degree) is 2, so we get -37, since in each combination there are 3 terms, we get (-1)^3= -1, and we multiply this by the -37 to get 37. If we add the rootsin this fashion manually, we can see that (1/2 * 2 *3)(1/2 * 2 * 4)(1/2 * 3 * 4)(2 * 3 * 4) = 37.