Expand Using Pascal's Triangle (x-y)^6

${(x - y)}^{6}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(x - y)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 6 - 15 - 20 - 15 - 6 - 1$ .

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $x$ and $b$ $- y$ into the expression.

$1 {(x)}^{6} {(- y)}^{0} + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Step 4

Simplify each term.

Multiply ${(x)}^{6}$ by $1$ .

${(x)}^{6} {(- y)}^{0} + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} ({(- 1)}^{0} y^{0}) + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} x^{6} y^{0} + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Anything raised to $0$ is $1$ .

$1 x^{6} y^{0} + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply $x^{6}$ by $1$ .

$x^{6} y^{0} + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Anything raised to $0$ is $1$ .

$x^{6} \cdot 1 + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply $x^{6}$ by $1$ .

$x^{6} + 6 {(x)}^{5} {(- y)}^{1} + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Simplify.

$x^{6} + 6 x^{5} (- y) + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} + 6 \cdot - 1 x^{5} y + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply $6$ by $- 1$ .

$x^{6} - 6 x^{5} y + 15 {(x)}^{4} {(- y)}^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} ({(- 1)}^{2} y^{2}) + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 \cdot {(- 1)}^{2} x^{4} y^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

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

$x^{6} - 6 x^{5} y + 15 \cdot 1 x^{4} y^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply $15$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 {(x)}^{3} {(- y)}^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 x^{3} ({(- 1)}^{3} y^{3}) + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 \cdot {(- 1)}^{3} x^{3} y^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

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

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} + 20 \cdot - 1 x^{3} y^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply $20$ by $- 1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 {(x)}^{2} {(- y)}^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} ({(- 1)}^{4} y^{4}) + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 \cdot {(- 1)}^{4} x^{2} y^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

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

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 \cdot 1 x^{2} y^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply $15$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 {(x)}^{1} {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Simplify.

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 x {(- y)}^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 x ({(- 1)}^{5} y^{5}) + 1 {(x)}^{0} {(- y)}^{6}$

Rewrite using the commutative property of multiplication.

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 \cdot {(- 1)}^{5} x y^{5} + 1 {(x)}^{0} {(- y)}^{6}$

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

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 \cdot - 1 x y^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply $6$ by $- 1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + 1 {(x)}^{0} {(- y)}^{6}$

Multiply ${(x)}^{0}$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + {(x)}^{0} {(- y)}^{6}$

Anything raised to $0$ is $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + 1 {(- y)}^{6}$

Multiply ${(- y)}^{6}$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + {(- y)}^{6}$

Apply the product rule to $- y$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + {(- 1)}^{6} y^{6}$

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

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + 1 y^{6}$

Multiply $y^{6}$ by $1$ .

$x^{6} - 6 x^{5} y + 15 x^{4} y^{2} - 20 x^{3} y^{3} + 15 x^{2} y^{4} - 6 x y^{5} + y^{6}$

Expand Using Pascal's Triangle (x-y)^6

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