Expand the Trigonometric Expression (2x-4)^4

${(2 x - 4)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $2 x$ .

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

Raise $2$ to the power of $4$ .

$16 x^{4} + 4 {(2 x)}^{3} \cdot - 4 + 6 {(2 x)}^{2} {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 4 (2^{3} x^{3}) \cdot - 4 + 6 {(2 x)}^{2} {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

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

$16 x^{4} + 4 (8 x^{3}) \cdot - 4 + 6 {(2 x)}^{2} {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Multiply $8$ by $4$ .

$16 x^{4} + 32 x^{3} \cdot - 4 + 6 {(2 x)}^{2} {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Multiply $- 4$ by $32$ .

$16 x^{4} - 128 x^{3} + 6 {(2 x)}^{2} {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} - 128 x^{3} + 6 (2^{2} x^{2}) {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

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

$16 x^{4} - 128 x^{3} + 6 (4 x^{2}) {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Multiply $4$ by $6$ .

$16 x^{4} - 128 x^{3} + 24 x^{2} {(- 4)}^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Raise $- 4$ to the power of $2$ .

$16 x^{4} - 128 x^{3} + 24 x^{2} \cdot 16 + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Multiply $16$ by $24$ .

$16 x^{4} - 128 x^{3} + 384 x^{2} + 4 (2 x) {(- 4)}^{3} + {(- 4)}^{4}$

Multiply $2$ by $4$ .

$16 x^{4} - 128 x^{3} + 384 x^{2} + 8 x {(- 4)}^{3} + {(- 4)}^{4}$

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

$16 x^{4} - 128 x^{3} + 384 x^{2} + 8 x \cdot - 64 + {(- 4)}^{4}$

Multiply $- 64$ by $8$ .

$16 x^{4} - 128 x^{3} + 384 x^{2} - 512 x + {(- 4)}^{4}$

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

$16 x^{4} - 128 x^{3} + 384 x^{2} - 512 x + 256$