Expand the Trigonometric Expression (3x-5)^4

${(3 x - 5)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $3 x$ .

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

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

$81 x^{4} + 4 {(3 x)}^{3} \cdot - 5 + 6 {(3 x)}^{2} {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Apply the product rule to $3 x$ .

$81 x^{4} + 4 (3^{3} x^{3}) \cdot - 5 + 6 {(3 x)}^{2} {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

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

$81 x^{4} + 4 (27 x^{3}) \cdot - 5 + 6 {(3 x)}^{2} {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Multiply $27$ by $4$ .

$81 x^{4} + 108 x^{3} \cdot - 5 + 6 {(3 x)}^{2} {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Multiply $- 5$ by $108$ .

$81 x^{4} - 540 x^{3} + 6 {(3 x)}^{2} {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Apply the product rule to $3 x$ .

$81 x^{4} - 540 x^{3} + 6 (3^{2} x^{2}) {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

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

$81 x^{4} - 540 x^{3} + 6 (9 x^{2}) {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Multiply $9$ by $6$ .

$81 x^{4} - 540 x^{3} + 54 x^{2} {(- 5)}^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Raise $- 5$ to the power of $2$ .

$81 x^{4} - 540 x^{3} + 54 x^{2} \cdot 25 + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Multiply $25$ by $54$ .

$81 x^{4} - 540 x^{3} + 1350 x^{2} + 4 (3 x) {(- 5)}^{3} + {(- 5)}^{4}$

Multiply $3$ by $4$ .

$81 x^{4} - 540 x^{3} + 1350 x^{2} + 12 x {(- 5)}^{3} + {(- 5)}^{4}$

Raise $- 5$ to the power of $3$ .

$81 x^{4} - 540 x^{3} + 1350 x^{2} + 12 x \cdot - 125 + {(- 5)}^{4}$

Multiply $- 125$ by $12$ .

$81 x^{4} - 540 x^{3} + 1350 x^{2} - 1500 x + {(- 5)}^{4}$

Raise $- 5$ to the power of $4$ .

$81 x^{4} - 540 x^{3} + 1350 x^{2} - 1500 x + 625$