Expand the Trigonometric Expression (3x-y)^7

${(3 x - y)}^{7}$

Step 1

Use the Binomial Theorem.

${(3 x)}^{7} + 7 {(3 x)}^{6} (- y) + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Step 2

Simplify each term.

Apply the product rule to $3 x$ .

$3^{7} x^{7} + 7 {(3 x)}^{6} (- y) + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Raise $3$ to the power of $7$ .

$2187 x^{7} + 7 {(3 x)}^{6} (- y) + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} + 7 \cdot - 1 {(3 x)}^{6} y + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $7$ by $- 1$ .

$2187 x^{7} - 7 {(3 x)}^{6} y + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 7 (3^{6} x^{6}) y + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Raise $3$ to the power of $6$ .

$2187 x^{7} - 7 (729 x^{6}) y + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $729$ by $- 7$ .

$2187 x^{7} - 5103 x^{6} y + 21 {(3 x)}^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 5103 x^{6} y + 21 (3^{5} x^{5}) {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 21 (243 x^{5}) {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $243$ by $21$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} {(- y)}^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $- y$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} ({(- 1)}^{2} y^{2}) + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 5103 x^{6} y + 5103 \cdot {(- 1)}^{2} x^{5} y^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 \cdot 1 x^{5} y^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $5103$ by $1$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} + 35 {(3 x)}^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} + 35 (3^{4} x^{4}) {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} + 35 (81 x^{4}) {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $81$ by $35$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} + 2835 x^{4} {(- y)}^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $- y$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} + 2835 x^{4} ({(- 1)}^{3} y^{3}) + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} + 2835 \cdot {(- 1)}^{3} x^{4} y^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} + 2835 \cdot - 1 x^{4} y^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $2835$ by $- 1$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 35 {(3 x)}^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 35 (3^{3} x^{3}) {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 35 (27 x^{3}) {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $27$ by $35$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} {(- y)}^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $- y$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} ({(- 1)}^{4} y^{4}) + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 \cdot {(- 1)}^{4} x^{3} y^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 \cdot 1 x^{3} y^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $945$ by $1$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} + 21 {(3 x)}^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $3 x$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} + 21 (3^{2} x^{2}) {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} + 21 (9 x^{2}) {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $9$ by $21$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} + 189 x^{2} {(- y)}^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $- y$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} + 189 x^{2} ({(- 1)}^{5} y^{5}) + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} + 189 \cdot {(- 1)}^{5} x^{2} y^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} + 189 \cdot - 1 x^{2} y^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $189$ by $- 1$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 7 (3 x) {(- y)}^{6} + {(- y)}^{7}$

Multiply $3$ by $7$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 21 x {(- y)}^{6} + {(- y)}^{7}$

Apply the product rule to $- y$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 21 x ({(- 1)}^{6} y^{6}) + {(- y)}^{7}$

Rewrite using the commutative property of multiplication.

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 21 \cdot {(- 1)}^{6} x y^{6} + {(- y)}^{7}$

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

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 21 \cdot 1 x y^{6} + {(- y)}^{7}$

Multiply $21$ by $1$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 21 x y^{6} + {(- y)}^{7}$

Apply the product rule to $- y$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 21 x y^{6} + {(- 1)}^{7} y^{7}$

Raise $- 1$ to the power of $7$ .

$2187 x^{7} - 5103 x^{6} y + 5103 x^{5} y^{2} - 2835 x^{4} y^{3} + 945 x^{3} y^{4} - 189 x^{2} y^{5} + 21 x y^{6} - y^{7}$

Expand the Trigonometric Expression (3x-y)^7

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