Simplify (x+2y+3z)^2

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

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

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

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

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

Simplify each term.

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Multiply $2$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $2$ by $3$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $z$ .

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

Multiply $z$ by $z$ .

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

Multiply $3$ by $3$ .

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

Add $2 x y$ and $2 y x$ .

Move $y$ .

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

Add $2 x y$ and $2 x y$ .

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

Add $3 x z$ and $3 z x$ .

Move $z$ .

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

Add $3 x z$ and $3 x z$ .

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

Add $6 y z$ and $6 z y$ .

Move $z$ .

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

Add $6 y z$ and $6 y z$ .

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

Move $12 y z$ .

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

Move $4 y^{2}$ .

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

Simplify (x+2y+3z)^2

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