Expand the Trigonometric Expression (x-y)^6

${(x - y)}^{6}$

Step 1

Use the Binomial Theorem.

$x^{6} + 6 x^{5} (- y) + 15 x^{4} {(- y)}^{2} + 20 x^{3} {(- y)}^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Step 2

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{6} + 6 \cdot - 1 x^{5} y + 15 x^{4} {(- y)}^{2} + 20 x^{3} {(- y)}^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Multiply $6$ by $- 1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} {(- y)}^{2} + 20 x^{3} {(- y)}^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} ({(- 1)}^{2} y^{2}) + 20 x^{3} {(- y)}^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 \cdot {(- 1)}^{2} x^{4} y^{2} + 20 x^{3} {(- y)}^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Raise $- 1$ to the power of $2$ .

$x^{6} - 6 x^{5} y + 15 \cdot 1 x^{4} y^{2} + 20 x^{3} {(- y)}^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Multiply $15$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 x^{3} {(- y)}^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 x^{3} ({(- 1)}^{3} y^{3}) + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 \cdot {(- 1)}^{3} x^{3} y^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Raise $- 1$ to the power of $3$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 \cdot - 1 x^{3} y^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Multiply $20$ by $- 1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} {(- y)}^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} ({(- 1)}^{4} y^{4}) + 6 x {(- y)}^{5} + {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 \cdot {(- 1)}^{4} x^{2} y^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Raise $- 1$ to the power of $4$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 \cdot 1 x^{2} y^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Multiply $15$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 x {(- y)}^{5} + {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 x ({(- 1)}^{5} y^{5}) + {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 \cdot {(- 1)}^{5} x y^{5} + {(- y)}^{6}$

Raise $- 1$ to the power of $5$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 \cdot - 1 x y^{5} + {(- y)}^{6}$

Multiply $6$ by $- 1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + {(- 1)}^{6} y^{6}$

Raise $- 1$ to the power of $6$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + 1 y^{6}$

Multiply $y^{6}$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + y^{6}$

Expand the Trigonometric Expression (x-y)^6

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