Question

If a + b + c = 0

Then prove that

a^{​​​​​​5} + b^{​​​​​​5} + c​​​​​^{​​​​​​5} a³ + b^{​​​​​​3} + c^{​​​​​​3} a^{​​​​​​2} + b^{​​​​​​2} + c^{​​​​​​2}

------------------- = ---------------------- × --------------------------

5 3 2

Answer 1

a+b+c=0 ; Squaring

(a+b+c)^2=0

a^2+b^2+c^2=−2(ab+bc+ca)……………(1)

(a+b)^3=a^3+b^3+3ab(a+b) but a+b=−ca+b=−c

−c^3=a^3+b^3−3abc or a^3+b^3+c^3=3abc…..(2)

Now (a^3+b^3+c^3(a^2+b^2+c^2)=a^5+b^5+c^5+a^3b^2+a^3c^2+b^3a^2+b^3c^2+c^3a^2+c^3b^2

=a^5+b^5+c^5+a^2b^2(a+b)+b^2c^2(b+c)+c^2a^2(c+a)

=a^5+b^5+c^5−a^2b^2c−b^2c^2a−c^2a^2b

=a^5+b^5+c^5−abc(ab+bc+ca)

So a^5+b^5+c^5=(a^3+b^3+c^3)(a^2+b^2+c^2)+abc(ab+bc+ca)

=(a^3+b^3+c^3)(a^2+b^2+c^2)+(a^3+b^3+c^3)/3*(a^2+b^2+c^2)/−2

=5/6(a^3+b^3+c^3)(a^2+b^2+c^2)

(a^5+b^5+c^5)/5=(a^3+b^3+c^3)/3*(a^2+b^2+c^2)/2