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

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

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

simplify

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

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

Solution steps

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

Expand

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

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

Expand

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

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

Expand

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

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

Group like terms

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

Add similar elements:

- 3 a b^{2} c + 3 a b^{2} c = 0

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

Add similar elements:

3 ab c^{2} - 3 ab c^{2} = 0

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

Add similar elements:

3 a^{2} b c - 3 a^{2} b c = 0

= a b^{3} - a c^{3} + b c^{3} - a^{3} b + a^{3} c - b^{3} c

\frac{\frac{a}{b}}{( 1 + \frac{a}{b} )}

\frac{m ^{3} n ^{- 5}}{m ^{- 2} n ^{- 2}}

(4 a^{4} + 7 b^{2})^{3}

f (arccos (\frac{1}{( 3 )}))

(10 - 6 x + x^{2})^{2}

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