Find the Product (3x+2y+z)^2

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $3$ .

$9 x^{2} + 3 x (2 y) + 3 x z + 2 y (3 x) + 2 y (2 y) + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} + 3 \cdot 2 x y + 3 x z + 2 y (3 x) + 2 y (2 y) + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Multiply $3$ by $2$ .

$9 x^{2} + 6 x y + 3 x z + 2 y (3 x) + 2 y (2 y) + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} + 6 x y + 3 x z + 2 \cdot 3 y x + 2 y (2 y) + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Multiply $2$ by $3$ .

$9 x^{2} + 6 x y + 3 x z + 6 y x + 2 y (2 y) + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} + 6 x y + 3 x z + 6 y x + 2 \cdot 2 y \cdot y + 2 y z + z (3 x) + z (2 y) + z \cdot z$

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

Move $y$ .

$9 x^{2} + 6 x y + 3 x z + 6 y x + 2 \cdot 2 (y \cdot y) + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Multiply $y$ by $y$ .

$9 x^{2} + 6 x y + 3 x z + 6 y x + 2 \cdot 2 y^{2} + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Multiply $2$ by $2$ .

$9 x^{2} + 6 x y + 3 x z + 6 y x + 4 y^{2} + 2 y z + z (3 x) + z (2 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} + 6 x y + 3 x z + 6 y x + 4 y^{2} + 2 y z + 3 z x + z (2 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

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

Multiply $z$ by $z$ .

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

Step 4

Add $6 x y$ and $6 y x$ .

Move $y$ .

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

Add $6 x y$ and $6 x y$ .

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

Step 5

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

Move $z$ .

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

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

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

Step 6

Add $2 y z$ and $2 z y$ .

Move $z$ .

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

Add $2 y z$ and $2 y z$ .

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

Find the Product (3x+2y+z)^2

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