Algebra Examples

{(x - y)}^{12}

${(x - y)}^{12}$

Step 1

Pascal's Triangle can be displayed as such:

1

$1$

1 - 1

$1 - 1$

1 - 2 - 1

$1 - 2 - 1$

\dots

$\dots$

1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1

$1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$

1 - 11 - 55 - 165 - 330 - 462 - 462 - 330 - 165 - 55 - 11 - 1

$1 - 11 - 55 - 165 - 330 - 462 - 462 - 330 - 165 - 55 - 11 - 1$

1 - 12 - 66 - 220 - 495 - 792 - 924 - 792 - 495 - 220 - 66 - 12 - 1

$1 - 12 - 66 - 220 - 495 - 792 - 924 - 792 - 495 - 220 - 66 - 12 - 1$

The triangle can be used to calculate the coefficients of the expansion of

{(a + b)}^{n}

${(a + b)}^{n}$ by taking the exponent

n

$n$ and adding

1

$1$ . The coefficients will correspond with line

n + 1

$n + 1$ of the triangle. For

{(x - y)}^{12}

${(x - y)}^{12}$ ,

n = 12

$n = 12$ so the coefficients of the expansion will correspond with line

13

$13$ .

Step 2

The expansion follows the rule

{(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}

${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are

1 - 12 - 66 - 220 - 495 - 792 - 924 - 792 - 495 - 220 - 66 - 12 - 1

$1 - 12 - 66 - 220 - 495 - 792 - 924 - 792 - 495 - 220 - 66 - 12 - 1$ .

1 a^{12} b^{0} + 12 a^{11} b + 66 a^{10} b^{2} + 220 a^{9} b^{3} + 495 a^{8} b^{4} + 792 a^{7} b^{5} + 924 a^{6} b^{6} + 792 a^{5} b^{7} + 495 a^{4} b^{8} + 220 a^{3} b^{9} + 66 a^{2} b^{10} + 12 a b^{11} + 1 a^{0} b^{12}

$1 a^{12} b^{0} + 12 a^{11} b + 66 a^{10} b^{2} + 220 a^{9} b^{3} + 495 a^{8} b^{4} + 792 a^{7} b^{5} + 924 a^{6} b^{6} + 792 a^{5} b^{7} + 495 a^{4} b^{8} + 220 a^{3} b^{9} + 66 a^{2} b^{10} + 12 a b^{11} + 1 a^{0} b^{12}$

Step 3

Substitute the actual values of

a

$a$

x

$x$ and

b

$b$

- y

$- y$ into the expression.

1 {(x)}^{12} {(- y)}^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$1 {(x)}^{12} {(- y)}^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply

{(x)}^{12}

${(x)}^{12}$ by

1

$1$ .

{(x)}^{12} {(- y)}^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

${(x)}^{12} {(- y)}^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.2

Apply the product rule to

- y

$- y$ .

x^{12} ({(- 1)}^{0} y^{0}) + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$x^{12} ({(- 1)}^{0} y^{0}) + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.3

Rewrite using the commutative property of multiplication.

{(- 1)}^{0} x^{12} y^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

${(- 1)}^{0} x^{12} y^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.4

Anything raised to

0

$0$ is

1

$1$ .

1 x^{12} y^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$1 x^{12} y^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.5

Multiply

x^{12}

$x^{12}$ by

1

$1$ .

x^{12} y^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$x^{12} y^{0} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.6

Anything raised to

0

$0$ is

1

$1$ .

x^{12} \cdot 1 + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$x^{12} \cdot 1 + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.7

Multiply

x^{12}

$x^{12}$ by

1

$1$ .

x^{12} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$x^{12} + 12 {(x)}^{11} {(- y)}^{1} + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.8

Simplify.

x^{12} + 12 x^{11} (- y) + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$x^{12} + 12 x^{11} (- y) + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.9

Rewrite using the commutative property of multiplication.

x^{12} + 12 \cdot - 1 x^{11} y + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}

$x^{12} + 12 \cdot - 1 x^{11} y + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.10

Multiply

12

$12$ by

$- 1$ .

$x^{12} - 12 x^{11} y + 66 {(x)}^{10} {(- y)}^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.11

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} ({(- 1)}^{2} y^{2}) + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.12

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 \cdot {(- 1)}^{2} x^{10} y^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.13

Raise

$- 1$ to the power of

$2$ .

$x^{12} - 12 x^{11} y + 66 \cdot 1 x^{10} y^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.14

Multiply

$66$ by

$1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} + 220 {(x)}^{9} {(- y)}^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.15

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} + 220 x^{9} ({(- 1)}^{3} y^{3}) + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.16

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} + 220 \cdot {(- 1)}^{3} x^{9} y^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.17

Raise

$- 1$ to the power of

$3$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} + 220 \cdot - 1 x^{9} y^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.18

Multiply

$220$ by

$- 1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 {(x)}^{8} {(- y)}^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.19

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} ({(- 1)}^{4} y^{4}) + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.20

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 \cdot {(- 1)}^{4} x^{8} y^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.21

Raise

$- 1$ to the power of

$4$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 \cdot 1 x^{8} y^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.22

Multiply

$495$ by

$1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 {(x)}^{7} {(- y)}^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.23

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 x^{7} ({(- 1)}^{5} y^{5}) + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.24

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 \cdot {(- 1)}^{5} x^{7} y^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.25

Raise

$- 1$ to the power of

$5$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 \cdot - 1 x^{7} y^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.26

Multiply

$792$ by

$- 1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 {(x)}^{6} {(- y)}^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.27

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} ({(- 1)}^{6} y^{6}) + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.28

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 \cdot {(- 1)}^{6} x^{6} y^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.29

Raise

$- 1$ to the power of

$6$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 \cdot 1 x^{6} y^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.30

Multiply

$924$ by

$1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 {(x)}^{5} {(- y)}^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.31

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 x^{5} ({(- 1)}^{7} y^{7}) + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.32

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 \cdot {(- 1)}^{7} x^{5} y^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.33

Raise

$- 1$ to the power of

$7$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 \cdot - 1 x^{5} y^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.34

Multiply

$792$ by

$- 1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 {(x)}^{4} {(- y)}^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.35

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} ({(- 1)}^{8} y^{8}) + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.36

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 \cdot {(- 1)}^{8} x^{4} y^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.37

Raise

$- 1$ to the power of

$8$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 \cdot 1 x^{4} y^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.38

Multiply

$495$ by

$1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 {(x)}^{3} {(- y)}^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.39

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 x^{3} ({(- 1)}^{9} y^{9}) + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.40

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 \cdot {(- 1)}^{9} x^{3} y^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.41

Raise

$- 1$ to the power of

$9$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 \cdot - 1 x^{3} y^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.42

Multiply

$220$ by

$- 1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 {(x)}^{2} {(- y)}^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.43

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 x^{2} ({(- 1)}^{10} y^{10}) + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.44

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 \cdot {(- 1)}^{10} x^{2} y^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.45

Raise

$- 1$ to the power of

$10$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 \cdot 1 x^{2} y^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.46

Multiply

$66$ by

$1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 {(x)}^{1} {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.47

Simplify.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 x {(- y)}^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.48

Apply the product rule to

$- y$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 x ({(- 1)}^{11} y^{11}) + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.49

Rewrite using the commutative property of multiplication.

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 \cdot {(- 1)}^{11} x y^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.50

Raise

$- 1$ to the power of

$11$ .

Step 4.51

Multiply

$12$ by

$- 1$ .

$x^{12} - 12 x^{11} y + 66 x^{10} y^{2} - 220 x^{9} y^{3} + 495 x^{8} y^{4} - 792 x^{7} y^{5} + 924 x^{6} y^{6} - 792 x^{5} y^{7} + 495 x^{4} y^{8} - 220 x^{3} y^{9} + 66 x^{2} y^{10} - 12 x y^{11} + 1 {(x)}^{0} {(- y)}^{12}$

Step 4.52

Multiply

${(x)}^{0}$ by

$1$ .

Step 4.53

Anything raised to

$0$ is

$1$ .

Step 4.54

Multiply

${(- y)}^{12}$ by

$1$ .

Step 4.55

Apply the product rule to

$- y$ .

Step 4.56

Raise

$- 1$ to the power of

$12$ .

Step 4.57

Multiply

$y^{12}$ by

$1$ .