Relation between roots and coefficients of general polynomial equation in one variable
Answers
This is how it goes.....
Let α,β,γ be the roots of x3+px2+qx+r=0. Then,writing the expression x3+px2+qx+r in terms of α, β and γ gives (x−α)(x−β)(x−γ).
∴ x3+px2+qx+r=(x−α)(x−β)(x−γ).
=(x2−[α+β]x+αβ)(x−γ)
=x3−(α+β)x2+αβx−γx2+(α+β)γx−αβγ
=x3−(α+β+γ)x2+(αβ+βγ+γα)x−αβγ
∴ equating coefficients
(a) α+β+γ=−p.
(b) αβ+βγ+γα=q.
(c) αβγ=−r.
This, of course, applies to a cubic equation. Let us extend this to a more general equation
In general, if α1, α2, α3,…,αn are the roots of the equation p0xn+p1xn−1+p2xn−2+⋯+pn−1x+pn=0 where (p0≠0) then
sum of the roots =−p1p0
sum of products of the roots, two at a time =p2p0
sum of products if the roots, three at a time =−p3p0
sum of products of the roots, n at a time = (−1)n,pnp0
I was able to understand the cubic equation's part but I am completely lost with the general part (i.e an nth degree polynomial). I am looking for a simpler explanation of what that means. However I understand that it can be used as formulas for finding the roots but i need to know how did he obtain the above formulas.
Answer:
Let us take the quadratic equation of the general form ax^2 + bx + c = 0 where a (≠ 0) is the coefficient of x^2, b the coefficient of x and c, the constant term