Expand the Trigonometric Expression (2x-y+3z)^2

${(2 x - y + 3 z)}^{2}$

Step 1

Rewrite ${(2 x - y + 3 z)}^{2}$ as $(2 x - y + 3 z) (2 x - y + 3 z)$ .

$(2 x - y + 3 z) (2 x - y + 3 z)$

Step 2

Expand $(2 x - y + 3 z) (2 x - y + 3 z)$ by multiplying each term in the first expression by each term in the second expression.

$2 x (2 x) + 2 x (- y) + 2 x (3 z) - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Step 3

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x (- y) + 2 x (3 z) - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x (- y) + 2 x (3 z) - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x (- y) + 2 x (3 z) - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $2$ by $2$ .

$4 x^{2} + 2 x (- y) + 2 x (3 z) - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} + 2 \cdot - 1 x y + 2 x (3 z) - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $2$ by $- 1$ .

$4 x^{2} - 2 x y + 2 x (3 z) - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} - 2 x y + 2 \cdot 3 x z - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $2$ by $3$ .

$4 x^{2} - 2 x y + 6 x z - y (2 x) - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} - 2 x y + 6 x z - 1 \cdot 2 y x - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $- 1$ by $2$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x - y (- y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} - 2 x y + 6 x z - 2 y x - 1 \cdot - 1 y \cdot y - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

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

Move $y$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x - 1 \cdot - 1 (y \cdot y) - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $y$ by $y$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x - 1 \cdot - 1 y^{2} - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $- 1$ by $- 1$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x + 1 y^{2} - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

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

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - y (3 z) + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 1 \cdot 3 y z + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Multiply $- 1$ by $3$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 3 z (2 x) + 3 z (- y) + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 3 \cdot 2 z x + 3 z (- y) + 3 z (3 z)$

Multiply $3$ by $2$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 6 z x + 3 z (- y) + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 6 z x + 3 \cdot - 1 z y + 3 z (3 z)$

Multiply $3$ by $- 1$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 6 z x - 3 z y + 3 z (3 z)$

Rewrite using the commutative property of multiplication.

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 6 z x - 3 z y + 3 \cdot 3 z \cdot z$

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

Move $z$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 6 z x - 3 z y + 3 \cdot 3 (z \cdot z)$

Multiply $z$ by $z$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 6 z x - 3 z y + 3 \cdot 3 z^{2}$

Multiply $3$ by $3$ .

$4 x^{2} - 2 x y + 6 x z - 2 y x + y^{2} - 3 y z + 6 z x - 3 z y + 9 z^{2}$

Step 4

Subtract $2 y x$ from $- 2 x y$ .

Move $y$ .

$4 x^{2} - 2 x y - 2 x y + 6 x z + y^{2} - 3 y z + 6 z x - 3 z y + 9 z^{2}$

Subtract $2 x y$ from $- 2 x y$ .

$4 x^{2} - 4 x y + 6 x z + y^{2} - 3 y z + 6 z x - 3 z y + 9 z^{2}$

Step 5

Add $6 x z$ and $6 z x$ .

Move $z$ .

$4 x^{2} - 4 x y + y^{2} - 3 y z + 6 x z + 6 x z - 3 z y + 9 z^{2}$

Add $6 x z$ and $6 x z$ .

$4 x^{2} - 4 x y + y^{2} - 3 y z + 12 x z - 3 z y + 9 z^{2}$

Step 6

Subtract $3 z y$ from $- 3 y z$ .

Move $z$ .

$4 x^{2} - 4 x y + y^{2} - 3 y z - 3 y z + 12 x z + 9 z^{2}$

Subtract $3 y z$ from $- 3 y z$ .

$4 x^{2} - 4 x y + y^{2} - 6 y z + 12 x z + 9 z^{2}$

Expand the Trigonometric Expression (2x-y+3z)^2

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