Expand the Trigonometric Expression (-p+q)^5

${(- p + q)}^{5}$

Step 1

Use the Binomial Theorem.

${(- p)}^{5} + 5 {(- p)}^{4} q + 10 {(- p)}^{3} q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

Step 2

Simplify each term.

Apply the product rule to $- p$ .

${(- 1)}^{5} p^{5} + 5 {(- p)}^{4} q + 10 {(- p)}^{3} q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

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

$- p^{5} + 5 {(- p)}^{4} q + 10 {(- p)}^{3} q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

Apply the product rule to $- p$ .

$- p^{5} + 5 ({(- 1)}^{4} p^{4}) q + 10 {(- p)}^{3} q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

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

$- p^{5} + 5 (1 p^{4}) q + 10 {(- p)}^{3} q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

Multiply $p^{4}$ by $1$ .

$- p^{5} + 5 p^{4} q + 10 {(- p)}^{3} q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

Apply the product rule to $- p$ .

$- p^{5} + 5 p^{4} q + 10 ({(- 1)}^{3} p^{3}) q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

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

$- p^{5} + 5 p^{4} q + 10 (- p^{3}) q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

Multiply $- 1$ by $10$ .

$- p^{5} + 5 p^{4} q - 10 p^{3} q^{2} + 10 {(- p)}^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

Apply the product rule to $- p$ .

$- p^{5} + 5 p^{4} q - 10 p^{3} q^{2} + 10 ({(- 1)}^{2} p^{2}) q^{3} + 5 (- p) q^{4} + q^{5}$

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

$- p^{5} + 5 p^{4} q - 10 p^{3} q^{2} + 10 (1 p^{2}) q^{3} + 5 (- p) q^{4} + q^{5}$

Multiply $p^{2}$ by $1$ .

$- p^{5} + 5 p^{4} q - 10 p^{3} q^{2} + 10 p^{2} q^{3} + 5 (- p) q^{4} + q^{5}$

Multiply $- 1$ by $5$ .

$- p^{5} + 5 p^{4} q - 10 p^{3} q^{2} + 10 p^{2} q^{3} - 5 p q^{4} + q^{5}$