Question

if a+b+c=0,then prove that a square(b+c) + b square ( c+a) +c square (a+b) +3abc =0​

Answer 1

hope this May help you

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Answer 2

a + b + c = 0

a + b = -c ... (i)

b + c = -a ... (ii)

c + a = -b ... (iii)

from (i)

(a + b)³ = (-c)³

a³ + b³ + 3ab(a + b) = -c³

a³ + b³ + 3ab(-c) = -c³ ... (from i)

a³ + b³ - 3abc = -c³

a³ + b³ + c³ = 3abc ... (iv)

now,

a²(b + c) + b²(c + a) + c²(a + b) + 3abc

= a²(-a) + b²(-b) + c²(-c) + 3abc ... (from i, ii, iii)

= - a³ - b³ - c³ + 3abc

= - (a³ + b³ + c³) + 3abc

= - 3abc + 3abc ... (from iv)

= 0

Hence, if a + b + c = 0,

then a²(b + c) + b²(c + a) + c²(a + b) + 3abc = 0

... Hence Proved!