Solve (a b c)^2-(b c-a)^2 (c a-b)^2-(a b-c)^2

Question

Simplify the expression

- 3 a^{2} b^{2} c^{2} - b^{2} c^{4} a^{2} + 2 b^{3} c^{3} a - b^{4} c^{2} + 2 b c^{3} a^{3} + 2 b^{3} c a - a^{4} c^{2} + 2 a^{3} c b - 2 a^{2} b^{2} + 2 ab c - c^{2}

Evaluate

(ab c)^{2} - (b c - a)^{2} (c a - b)^{2} - (ab - c)^{2}

Rewrite the expression

a^{2} b^{2} c^{2} - (b c - a)^{2} (c a - b)^{2} - (ab - c)^{2}

Expand the expression

More Steps

Calculate

- (b c - a)^{2} (c a - b)^{2}

Simplify

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

Simplify

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

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

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

Apply the distributive property

- b^{2} c^{2} \times c^{2} a^{2} - (- b^{2} c^{2} \times 2 c ab) - b^{2} c^{2} b^{2} + 2 b c a c^{2} a^{2} - 2 b c a \times 2 c ab + 2 b c a b^{2} - a^{2} c^{2} a^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- b^{2} c^{2} \times 2 c ab) - b^{2} c^{2} b^{2} + 2 b c a c^{2} a^{2} - 2 b c a \times 2 c ab + 2 b c a b^{2} - a^{2} c^{2} a^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- 2 b^{3} c^{3} a) - b^{2} c^{2} b^{2} + 2 b c a c^{2} a^{2} - 2 b c a \times 2 c ab + 2 b c a b^{2} - a^{2} c^{2} a^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- 2 b^{3} c^{3} a) - b^{4} c^{2} + 2 b c a c^{2} a^{2} - 2 b c a \times 2 c ab + 2 b c a b^{2} - a^{2} c^{2} a^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- 2 b^{3} c^{3} a) - b^{4} c^{2} + 2 b c^{3} a^{3} - 2 b c a \times 2 c ab + 2 b c a b^{2} - a^{2} c^{2} a^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- 2 b^{3} c^{3} a) - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b c a b^{2} - a^{2} c^{2} a^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- 2 b^{3} c^{3} a) - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b^{3} c a - a^{2} c^{2} a^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- 2 b^{3} c^{3} a) - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b^{3} c a - a^{4} c^{2} - (- a^{2} \times 2 c ab) - a^{2} b^{2}

Multiply the terms

More Steps

- b^{2} c^{4} a^{2} - (- 2 b^{3} c^{3} a) - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b^{3} c a - a^{4} c^{2} - (- 2 a^{3} c b) - a^{2} b^{2}

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

- b^{2} c^{4} a^{2} + 2 b^{3} c^{3} a - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b^{3} c a - a^{4} c^{2} + 2 a^{3} c b - a^{2} b^{2}

a^{2} b^{2} c^{2} - b^{2} c^{4} a^{2} + 2 b^{3} c^{3} a - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b^{3} c a - a^{4} c^{2} + 2 a^{3} c b - a^{2} b^{2} - (ab - c)^{2}

Expand the expression

a^{2} b^{2} c^{2} - b^{2} c^{4} a^{2} + 2 b^{3} c^{3} a - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b^{3} c a - a^{4} c^{2} + 2 a^{3} c b - a^{2} b^{2} - (a^{2} b^{2} - 2 ab c + c^{2})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

a^{2} b^{2} c^{2} - b^{2} c^{4} a^{2} + 2 b^{3} c^{3} a - b^{4} c^{2} + 2 b c^{3} a^{3} - 4 b^{2} c^{2} a^{2} + 2 b^{3} c a - a^{4} c^{2} + 2 a^{3} c b - a^{2} b^{2} - a^{2} b^{2} + 2 ab c - c^{2}

Subtract the terms

More Steps

- 3 a^{2} b^{2} c^{2} - b^{2} c^{4} a^{2} + 2 b^{3} c^{3} a - b^{4} c^{2} + 2 b c^{3} a^{3} + 2 b^{3} c a - a^{4} c^{2} + 2 a^{3} c b - a^{2} b^{2} - a^{2} b^{2} + 2 ab c - c^{2}

Solution

More Steps

- 3 a^{2} b^{2} c^{2} - b^{2} c^{4} a^{2} + 2 b^{3} c^{3} a - b^{4} c^{2} + 2 b c^{3} a^{3} + 2 b^{3} c a - a^{4} c^{2} + 2 a^{3} c b - 2 a^{2} b^{2} + 2 ab c - c^{2}

Show Solution