Expand the Trigonometric Expression (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.

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

Step 3

Simplify each term.

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Rewrite using the commutative property of multiplication.

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

Multiply $z$ by $z$ .

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

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

Add $x z$ and $z x$ .

Reorder $z$ and $x$ .

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

Add $x z$ and $x z$ .

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

Step 6

Subtract $z y$ from $- y z$ .

Move $z$ .

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

Subtract $y z$ from $- y z$ .

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

Expand the Trigonometric Expression (x-y+z)^2

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