Precalculus Examples

${(2 x - y)}^{5}$

Step 1

Use the Binomial Theorem.

${(2 x)}^{5} + 5 {(2 x)}^{4} (- y) + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $2 x$ .

$2^{5} x^{5} + 5 {(2 x)}^{4} (- y) + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.2

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

$32 x^{5} + 5 {(2 x)}^{4} (- y) + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.3

Rewrite using the commutative property of multiplication.

$32 x^{5} + 5 \cdot - 1 {(2 x)}^{4} y + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.4

Multiply $5$ by $- 1$ .

$32 x^{5} - 5 {(2 x)}^{4} y + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.5

Apply the product rule to $2 x$ .

$32 x^{5} - 5 (2^{4} x^{4}) y + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.6

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

$32 x^{5} - 5 (16 x^{4}) y + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.7

Multiply $16$ by $- 5$ .

$32 x^{5} - 80 x^{4} y + 10 {(2 x)}^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.8

Apply the product rule to $2 x$ .

$32 x^{5} - 80 x^{4} y + 10 (2^{3} x^{3}) {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.9

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

$32 x^{5} - 80 x^{4} y + 10 (8 x^{3}) {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.10

Multiply $8$ by $10$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} {(- y)}^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.11

Apply the product rule to $- y$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} ({(- 1)}^{2} y^{2}) + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.12

Rewrite using the commutative property of multiplication.

$32 x^{5} - 80 x^{4} y + 80 \cdot {(- 1)}^{2} x^{3} y^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.13

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

$32 x^{5} - 80 x^{4} y + 80 \cdot 1 x^{3} y^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.14

Multiply $80$ by $1$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} + 10 {(2 x)}^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.15

Apply the product rule to $2 x$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} + 10 (2^{2} x^{2}) {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.16

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

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} + 10 (4 x^{2}) {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.17

Multiply $4$ by $10$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} + 40 x^{2} {(- y)}^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.18

Apply the product rule to $- y$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} + 40 x^{2} ({(- 1)}^{3} y^{3}) + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.19

Rewrite using the commutative property of multiplication.

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} + 40 \cdot {(- 1)}^{3} x^{2} y^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.20

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

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} + 40 \cdot - 1 x^{2} y^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.21

Multiply $40$ by $- 1$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 5 (2 x) {(- y)}^{4} + {(- y)}^{5}$

Step 2.22

Multiply $2$ by $5$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 10 x {(- y)}^{4} + {(- y)}^{5}$

Step 2.23

Apply the product rule to $- y$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 10 x ({(- 1)}^{4} y^{4}) + {(- y)}^{5}$

Step 2.24

Rewrite using the commutative property of multiplication.

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 10 \cdot {(- 1)}^{4} x y^{4} + {(- y)}^{5}$

Step 2.25

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

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 10 \cdot 1 x y^{4} + {(- y)}^{5}$

Step 2.26

Multiply $10$ by $1$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 10 x y^{4} + {(- y)}^{5}$

Step 2.27

Apply the product rule to $- y$ .

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 10 x y^{4} + {(- 1)}^{5} y^{5}$

Step 2.28

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

$32 x^{5} - 80 x^{4} y + 80 x^{3} y^{2} - 40 x^{2} y^{3} + 10 x y^{4} - y^{5}$