Expand the Trigonometric Expression (z-k)^6

${(z - k)}^{6}$

Step 1

Use the Binomial Theorem.

$z^{6} + 6 z^{5} (- k) + 15 z^{4} {(- k)}^{2} + 20 z^{3} {(- k)}^{3} + 15 z^{2} {(- k)}^{4} + 6 z {(- k)}^{5} + {(- k)}^{6}$

Step 2

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $6$ by $- 1$ .

$z^{6} - 6 z^{5} k + 15 z^{4} {(- k)}^{2} + 20 z^{3} {(- k)}^{3} + 15 z^{2} {(- k)}^{4} + 6 z {(- k)}^{5} + {(- k)}^{6}$

Apply the product rule to $- k$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $15$ by $1$ .

$z^{6} - 6 z^{5} k + 15 z^{4} k^{2} + 20 z^{3} {(- k)}^{3} + 15 z^{2} {(- k)}^{4} + 6 z {(- k)}^{5} + {(- k)}^{6}$

Apply the product rule to $- k$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $20$ by $- 1$ .

$z^{6} - 6 z^{5} k + 15 z^{4} k^{2} - 20 z^{3} k^{3} + 15 z^{2} {(- k)}^{4} + 6 z {(- k)}^{5} + {(- k)}^{6}$

Apply the product rule to $- k$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $15$ by $1$ .

$z^{6} - 6 z^{5} k + 15 z^{4} k^{2} - 20 z^{3} k^{3} + 15 z^{2} k^{4} + 6 z {(- k)}^{5} + {(- k)}^{6}$

Apply the product rule to $- k$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $6$ by $- 1$ .

$z^{6} - 6 z^{5} k + 15 z^{4} k^{2} - 20 z^{3} k^{3} + 15 z^{2} k^{4} - 6 z k^{5} + {(- k)}^{6}$

Apply the product rule to $- k$ .

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

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

$z^{6} - 6 z^{5} k + 15 z^{4} k^{2} - 20 z^{3} k^{3} + 15 z^{2} k^{4} - 6 z k^{5} + 1 k^{6}$

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

$z^{6} - 6 z^{5} k + 15 z^{4} k^{2} - 20 z^{3} k^{3} + 15 z^{2} k^{4} - 6 z k^{5} + k^{6}$

Expand the Trigonometric Expression (z-k)^6

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