Expand the Trigonometric Expression (3x-4y+z)^2

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

Step 1

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

$(3 x - 4 y + z) (3 x - 4 y + z)$

Step 2

Expand $(3 x - 4 y + z) (3 x - 4 y + z)$ by multiplying each term in the first expression by each term in the second expression.

$3 x (3 x) + 3 x (- 4 y) + 3 x z - 4 y (3 x) - 4 y (- 4 y) - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Step 3

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 \cdot 3 x \cdot x + 3 x (- 4 y) + 3 x z - 4 y (3 x) - 4 y (- 4 y) - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

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

Move $x$ .

$3 \cdot 3 (x \cdot x) + 3 x (- 4 y) + 3 x z - 4 y (3 x) - 4 y (- 4 y) - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Multiply $x$ by $x$ .

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

Multiply $3$ by $3$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $- 4$ .

$9 x^{2} - 12 x y + 3 x z - 4 y (3 x) - 4 y (- 4 y) - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} - 12 x y + 3 x z - 4 \cdot 3 y x - 4 y (- 4 y) - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Multiply $- 4$ by $3$ .

$9 x^{2} - 12 x y + 3 x z - 12 y x - 4 y (- 4 y) - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} - 12 x y + 3 x z - 12 y x - 4 \cdot - 4 y \cdot y - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

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

Move $y$ .

$9 x^{2} - 12 x y + 3 x z - 12 y x - 4 \cdot - 4 (y \cdot y) - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Multiply $y$ by $y$ .

$9 x^{2} - 12 x y + 3 x z - 12 y x - 4 \cdot - 4 y^{2} - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Multiply $- 4$ by $- 4$ .

$9 x^{2} - 12 x y + 3 x z - 12 y x + 16 y^{2} - 4 y z + z (3 x) + z (- 4 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} - 12 x y + 3 x z - 12 y x + 16 y^{2} - 4 y z + 3 z x + z (- 4 y) + z \cdot z$

Rewrite using the commutative property of multiplication.

$9 x^{2} - 12 x y + 3 x z - 12 y x + 16 y^{2} - 4 y z + 3 z x - 4 z y + z \cdot z$

Multiply $z$ by $z$ .

$9 x^{2} - 12 x y + 3 x z - 12 y x + 16 y^{2} - 4 y z + 3 z x - 4 z y + z^{2}$

Step 4

Subtract $12 y x$ from $- 12 x y$ .

Move $y$ .

$9 x^{2} - 12 x y - 12 x y + 3 x z + 16 y^{2} - 4 y z + 3 z x - 4 z y + z^{2}$

Subtract $12 x y$ from $- 12 x y$ .

$9 x^{2} - 24 x y + 3 x z + 16 y^{2} - 4 y z + 3 z x - 4 z y + z^{2}$

Step 5

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

Move $z$ .

$9 x^{2} - 24 x y + 16 y^{2} - 4 y z + 3 x z + 3 x z - 4 z y + z^{2}$

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

$9 x^{2} - 24 x y + 16 y^{2} - 4 y z + 6 x z - 4 z y + z^{2}$

Step 6

Subtract $4 z y$ from $- 4 y z$ .

Move $z$ .

$9 x^{2} - 24 x y + 16 y^{2} - 4 y z - 4 y z + 6 x z + z^{2}$

Subtract $4 y z$ from $- 4 y z$ .

$9 x^{2} - 24 x y + 16 y^{2} - 8 y z + 6 x z + z^{2}$

Expand the Trigonometric Expression (3x-4y+z)^2

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