Simplify a(b+c-bc)-b(c+a-ac)+c(b-a)

$a (b + c - b c) - b (c + a - a c) + c (b - a)$

Step 1

Simplify each term.

Apply the distributive property.

$a b + a c + a (- b c) - b (c + a - a c) + c (b - a)$

Rewrite using the commutative property of multiplication.

$a b + a c - a b c - b (c + a - a c) + c (b - a)$

Apply the distributive property.

$a b + a c - a b c - b c - b a - b (- a c) + c (b - a)$

Multiply $- b (- a c)$ .

Multiply $- 1$ by $- 1$ .

$a b + a c - a b c - b c - b a + 1 b (a c) + c (b - a)$

Multiply $b$ by $1$ .

$a b + a c - a b c - b c - b a + b (a c) + c (b - a)$

Apply the distributive property.

$a b + a c - a b c - b c - b a + b a c + c b + c (- a)$

Rewrite using the commutative property of multiplication.

$a b + a c - a b c - b c - b a + b a c + c b - c a$

Step 2

Combine the opposite terms in $a b + a c - a b c - b c - b a + b a c + c b - c a$ .

Reorder the factors in the terms $a b$ and $- b a$ .

$a b + a c - a b c - b c - a b + b a c + c b - c a$

Subtract $a b$ from $a b$ .

$a c - a b c - b c + 0 + b a c + c b - c a$

Add $a c - a b c - b c$ and $0$ .

$a c - a b c - b c + b a c + c b - c a$

Reorder the factors in the terms $- a b c$ and $b a c$ .

$a c - a b c - b c + a b c + c b - c a$

Add $- a b c$ and $a b c$ .

$a c - b c + 0 + c b - c a$

Add $a c - b c$ and $0$ .

$a c - b c + c b - c a$

Reorder the factors in the terms $- b c$ and $c b$ .

$a c - b c + b c - c a$

Add $- b c$ and $b c$ .

$a c + 0 - c a$

Add $a c$ and $0$ .

$a c - c a$

Reorder the factors in the terms $a c$ and $- c a$ .

$a c - a c$

Subtract $a c$ from $a c$ .

$0$