Expand the Trigonometric Expression (3y-1)^4

${(3 y - 1)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $3 y$ .

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

Raise $3$ to the power of $4$ .

$81 y^{4} + 4 {(3 y)}^{3} \cdot - 1 + 6 {(3 y)}^{2} {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Apply the product rule to $3 y$ .

$81 y^{4} + 4 (3^{3} y^{3}) \cdot - 1 + 6 {(3 y)}^{2} {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

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

$81 y^{4} + 4 (27 y^{3}) \cdot - 1 + 6 {(3 y)}^{2} {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Multiply $27$ by $4$ .

$81 y^{4} + 108 y^{3} \cdot - 1 + 6 {(3 y)}^{2} {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Multiply $- 1$ by $108$ .

$81 y^{4} - 108 y^{3} + 6 {(3 y)}^{2} {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Apply the product rule to $3 y$ .

$81 y^{4} - 108 y^{3} + 6 (3^{2} y^{2}) {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Raise $3$ to the power of $2$ .

$81 y^{4} - 108 y^{3} + 6 (9 y^{2}) {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Multiply $9$ by $6$ .

$81 y^{4} - 108 y^{3} + 54 y^{2} {(- 1)}^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

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

$81 y^{4} - 108 y^{3} + 54 y^{2} \cdot 1 + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Multiply $54$ by $1$ .

$81 y^{4} - 108 y^{3} + 54 y^{2} + 4 (3 y) {(- 1)}^{3} + {(- 1)}^{4}$

Multiply $3$ by $4$ .

$81 y^{4} - 108 y^{3} + 54 y^{2} + 12 y {(- 1)}^{3} + {(- 1)}^{4}$

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

$81 y^{4} - 108 y^{3} + 54 y^{2} + 12 y \cdot - 1 + {(- 1)}^{4}$

Multiply $- 1$ by $12$ .

$81 y^{4} - 108 y^{3} + 54 y^{2} - 12 y + {(- 1)}^{4}$

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

$81 y^{4} - 108 y^{3} + 54 y^{2} - 12 y + 1$