## alexd00d 3 years ago 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

1. alexd00d

cubic polynominals

2. liza2

I have no idea, I'm commenting so that I'll get a notification when someone solves this.

3. alexd00d

If someone gets this they are the smartest person I know!

4. joemath314159

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.

5. joemath314159

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 $a= 2, b=1-\sqrt{2}, c=1+\sqrt{2}$

6. joemath314159

mistype again, +2 instead of -2 >.>

7. alexd00d

You must have had loads of practice to be as good as you are!