Simplify (x-y-z)^2

${(x - y - z)}^{2}$

Step 1

Rewrite ${(x - y - z)}^{2}$ as $(x - y - z) (x - y - z)$ .

$(x - y - z) (x - y - z)$

Step 2

Expand $(x - y - z) (x - y - z)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x (- y) + x (- z) - y x - y (- y) - y (- z) - z x - z (- y) - z (- z)$

Step 3

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (- y) + x (- z) - y x - y (- y) - y (- z) - z x - z (- y) - z (- z)$

Rewrite using the commutative property of multiplication.

$x^{2} - x y + x (- z) - y x - y (- y) - y (- z) - z x - z (- y) - z (- z)$

Rewrite using the commutative property of multiplication.

$x^{2} - x y - x z - y x - y (- y) - y (- z) - z x - z (- y) - z (- z)$

Rewrite using the commutative property of multiplication.

$x^{2} - x y - x z - y x - 1 \cdot - 1 y \cdot y - y (- z) - z x - z (- y) - z (- z)$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$x^{2} - x y - x z - y x - 1 \cdot - 1 (y \cdot y) - y (- z) - z x - z (- y) - z (- z)$

Multiply $y$ by $y$ .

$x^{2} - x y - x z - y x - 1 \cdot - 1 y^{2} - y (- z) - z x - z (- y) - z (- z)$

Multiply $- 1$ by $- 1$ .

$x^{2} - x y - x z - y x + 1 y^{2} - y (- z) - z x - z (- y) - z (- z)$

Multiply $y^{2}$ by $1$ .

$x^{2} - x y - x z - y x + y^{2} - y (- z) - z x - z (- y) - z (- z)$

Rewrite using the commutative property of multiplication.

$x^{2} - x y - x z - y x + y^{2} - 1 \cdot - 1 y z - z x - z (- y) - z (- z)$

Multiply $- 1$ by $- 1$ .

$x^{2} - x y - x z - y x + y^{2} + 1 y z - z x - z (- y) - z (- z)$

Multiply $y$ by $1$ .

$x^{2} - x y - x z - y x + y^{2} + y z - z x - z (- y) - z (- z)$

Rewrite using the commutative property of multiplication.

$x^{2} - x y - x z - y x + y^{2} + y z - z x - 1 \cdot - 1 z y - z (- z)$

Multiply $- 1$ by $- 1$ .

$x^{2} - x y - x z - y x + y^{2} + y z - z x + 1 z y - z (- z)$

Multiply $z$ by $1$ .

$x^{2} - x y - x z - y x + y^{2} + y z - z x + z y - z (- z)$

Rewrite using the commutative property of multiplication.

$x^{2} - x y - x z - y x + y^{2} + y z - z x + z y - 1 \cdot - 1 z \cdot z$

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

$x^{2} - x y - x z - y x + y^{2} + y z - z x + z y - 1 \cdot - 1 (z \cdot z)$

Multiply $z$ by $z$ .

$x^{2} - x y - x z - y x + y^{2} + y z - z x + z y - 1 \cdot - 1 z^{2}$

Multiply $- 1$ by $- 1$ .

$x^{2} - x y - x z - y x + y^{2} + y z - z x + z y + 1 z^{2}$

Multiply $z^{2}$ by $1$ .

$x^{2} - x y - x z - y x + y^{2} + y z - z x + z y + z^{2}$

Step 4

Subtract $y x$ from $- x y$ .

Move $y$ .

$x^{2} - x y - 1 x y - x z + y^{2} + y z - z x + z y + z^{2}$

Subtract $x y$ from $- x y$ .

$x^{2} - 2 x y - x z + y^{2} + y z - z x + z y + z^{2}$

Step 5

Subtract $z x$ from $- x z$ .

Move $z$ .

$x^{2} - 2 x y + y^{2} + y z - x z - 1 x z + z y + z^{2}$

Subtract $x z$ from $- x z$ .

$x^{2} - 2 x y + y^{2} + y z - 2 x z + z y + z^{2}$

Step 6

Add $y z$ and $z y$ .

Reorder $z$ and $y$ .

$x^{2} - 2 x y + y^{2} + y z + y z - 2 x z + z^{2}$

Add $y z$ and $y z$ .

$x^{2} - 2 x y + y^{2} + 2 y z - 2 x z + z^{2}$

Step 7

Move $2 y z$ .

$x^{2} - 2 x y + y^{2} - 2 x z + 2 y z + z^{2}$

Step 8

Move $y^{2}$ .

$x^{2} - 2 x y - 2 x z + y^{2} + 2 y z + z^{2}$

Simplify (x-y-z)^2

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