What is (a+b+c)^3-a^3-b^3-c^3 ?

The solution to (a+b+c)^3-a^3-b^3-c^3 is 3a^2b+3a^2c+3ac^2+3bc^2+3ab^2+3b^2c+6abc

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simplify (a+b+c)^3-a^3-b^3-c^3

simplify

(a + b + c)^{3} - a^{3} - b^{3} - c^{3}

3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + 3 b^{2} c + 6 ab c

Solution steps

(a + b + c)^{3} - a^{3} - b^{3} - c^{3}

Expand

(a + b + c)^{3} : a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + c^{3} + 3 b^{2} c + 6 ab c + b^{3}

= a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + c^{3} + 3 b^{2} c + 6 ab c + b^{3} - a^{3} - b^{3} - c^{3}

Group like terms

= a^{3} - a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + c^{3} - c^{3} + 3 b^{2} c + 6 ab c + b^{3} - b^{3}

Add similar elements:

a^{3} - a^{3} = 0

= 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + c^{3} - c^{3} + 3 b^{2} c + 6 ab c + b^{3} - b^{3}

Add similar elements:

c^{3} - c^{3} = 0

= 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + 3 b^{2} c + 6 ab c + b^{3} - b^{3}

Add similar elements:

b^{3} - b^{3} = 0

= 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + 3 b^{2} c + 6 ab c

9 x^{3} y + 41 x^{2} y + 20 x y^{3}

- a^{4} + 4 b^{2}

- 2 x (x + 3) - (x + 1) (x - 2)

(x^{- 1} \cdot y^{3})^{\frac{1}{2}}

\frac{2 ^{3} ( x ^{2} ) ^{3}}{2 ^{2}}

What is (a+b+c)^3-a^3-b^3-c^3 ?
The solution to (a+b+c)^3-a^3-b^3-c^3 is 3a^2b+3a^2c+3ac^2+3bc^2+3ab^2+3b^2c+6abc