This question was uploaded on 30/10/21 on social media accounts.
Three higher degree equations on three variables. The Sum of three variables, the sum of the squares of three variables, and the sum of the cubes of three variables are given. Find those three variables.
Solution:
Let,
\[a\ge b\ge c\]
\[a+b+c=3----(1)\]
\[a^2+b^2+c^2=5----(2)\]
\[a^3+b^3+c^3=9----(3)\]
We know that:
\[a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)\]
\[\Rightarrow ab+bc+ca=2----(4)\]
Also,
\[a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\]
\[\Rightarrow abc=0----(5)\]
From the above equation, at least one element is zero.
If a is zero then b,c are negative (Equation (1) doesn't satisfy).
If b is zero then a,c are positive, negative (Equation (4) doesn't satisfy).
c must be zero:
Now,
\[c=0\]
By equation (4),
\[ab=2\]
By equation (1),
\[a+b=3\]
\[\Rightarrow a=2,⠀b=1\]
And, {a, b, c} = {0, 1, 2}
