Question

IF A+B+C=0 THEN A4+B4+C4/A2B2+B2C2+C2A2=

Answer 1

it would equal zero then.

Answer 2

That is if A B and C are equal to zero

Answer 3

note that\[(x+y+z)^2=x^2+y^2+z^2+2(xy+xz+yz)\]so\[x^2+y^2+z^2=(x+y+z)^2-2(xy+xz+yz) \ \ \ (\star)\]now let \(x=a^2\) , \(y=b^2\) and \(z=c^2\) in \((\star)\)\[a^4+b^4+c^4=(\color\red{a^2+b^2+c^2})^2-2(a^2b^2+a^2c^2+b^2c^2)\]\[a^4+b^4+c^4=(\color\red{(a+b+c)^2-2(ab+ac+bc)})^2-2(a^2b^2+a^2c^2+b^2c^2)\]we know that \(a+b+c=0\) the expression becomes\[a^4+b^4+c^4=(\color\red{-2(ab+ac+bc)})^2-2(a^2b^2+a^2c^2+b^2c^2)\]\[a^4+b^4+c^4=\color\red{4(a^2b^2+a^2c^2+b^2c^2)+8abc(a+b+c)}-2(a^2b^2+a^2c^2+b^2c^2)\]using \(a+b+c=0\) again\[a^4+b^4+c^4=2(a^2b^2+a^2c^2+b^2c^2)\]we are done