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a+b+c=4 a^2+b^2+c^2=10 a^3+b^3+c^3=22 find the values of a, b ,and c

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cubic polynominals
I have no idea, I'm commenting so that I'll get a notification when someone solves this.
If someone gets this they are the smartest person I know!

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Other answers:

First note that:\[(a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc)\Longrightarrow \]\[4^2=10+2(ab+ac+bc)\Longrightarrow ab+ac+bc=3\]Also:\[(a+b+c)^3=a^3+b^3+c^3+3(a+b+c)(ab+ac+bc)-3abc\Longrightarrow\]\[4^3=22+3(4)(3)-3abc\Longrightarrow abc=-2\]So your system is equivalent to: a+b+c=10 ab+ac+bc=3 abc=-2 Solving this system is the same as solving the polynomial: \[x^3-10x^2+3x+2=0\]So the solutions to this polynomial are the solutions to your system.
oh, mistype, thats why things were acting weird. Your system is equivalent to:\[a+b+c=4\]\[ab+ac+bc=3\]\[abc=-2\]which gives the polynomial:\[x^3-4x+3x-2=0\]which gives solutions \[a= 2, b=1-\sqrt{2}, c=1+\sqrt{2}\]
mistype again, +2 instead of -2 >.>
You must have had loads of practice to be as good as you are!

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