Question

what is the formula of (a+b+c)³?

Answer 1

The formula of (a+b+c)^3 = a^3+ b^3+ c^3 +3 (a+b) (b+c) (c+a)

Proof:
(a+b+c)^3= a^3+ b^3+ c^3 +3 (a+b) (b+c) (c+a)

it can be written as
(a+b+c)^3 - a^3 - b^3 - c^3 = 3 (a+b) (b+c) (c+a) .............(1)

Consider the L.H.S of equation (1)

(a+b+c)^3 - a^3 - b^3 - c^3 = a^3 + b^3 + c^3 +3ab (a+b) +3bc (b+c) +3ca (c+a) + 6abc - a^3 - b^3 - c^3

= 3ab (a+b) +3bc (b+c) +3ca (c+a) + 6abc

= 3[ ab (a+b) +bc (b+c) +ca (c+a) +2abc]

= 3[ab (a+b) + b^2c + bc^2 + abc + a^2c + ac^2 + abc]

= 3[ab (a+b) + (abc+ b^2c) + (abc+ a^2c) + (bc^2+ ac^2]

= 3[ (a+b) (ab+bc+ca+c^2)]

= 3[ (a+b) {(c^2+ bc) +(ab+ ac)}]

= 3[ (a+b) { c(b+c) +a(b+c)}]

= 3 (a+b) (b+c) (c+a)

which is equal to R.H.S of equation (1)

Hence proved

HOPE it helps!!!!