Relationship between roots and coefficients for 4th Order Polynomials

If the equation ax^4 + bx^3 + cx^2 + dx+ e = 0 has the roots alpha α, beta β, gamma ɣ, and delta δ, then was are the relationships between these roots and the coefficients a, b, c and d? We find out in this video and we apply these to the following example:

We solve for α, β, ɣ, and δ given the equation x^4 - 2x^3 +4x^2 +6x - 21 = 0 and α + β = 0.

Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

[hyperduality]

Complex roots come in pairs -- "Sith lords come in pairs (duals)" -- Obi Wan Kenobi.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.

[migo]

Hi, In your video you solved u & v by inspection and you found u = -3 & v = 7. That's fine with me but when I let u = 7 and v = -3 (this should still satisfy the equations u+v = 4 & uv = -21) and continued the problem, I found that I ended up with 2 complex conjugate roots being sqrt(7) * i & - sqrt(7) * i which doesn't match with yours. Why?

[rayzard]

Hlo im new
News:2020-pandemic covid, lockdown, most memes and we are here

[mamadetaslimtorabally]

Good but quite a lengthy process. Unfortunately, we have to live with it and master it through practice until the whole process becomes a natural way with us solving these types of quartic equations.

[dewangsingh]

a good solution but we can also do it in an easier way without remembering the formulas mentioned.
just put alpha in the equation and then put beta as -ve of alpha in the same equation separately and subtract these two equations..
you will get alpha and beta by that so now you have two roots just multiply them together because that will also be the factor of the original equation and divide it from original one..
the resultant quotient will give you the other two..