Expand Using Pascal's Triangle (s-v)^6

${(s - v)}^{6}$

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 ${(s - v)}^{6}$ , $n = 6$ 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 $a$ $s$ and $b$ $- v$ into the expression.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $- v$ .

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

Rewrite using the commutative property of multiplication.

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

Anything raised to $0$ is $1$ .

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

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

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

Anything raised to $0$ is $1$ .

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

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

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Multiply $6$ by $- 1$ .

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

Apply the product rule to $- v$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $15$ by $1$ .

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

Apply the product rule to $- v$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $20$ by $- 1$ .

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

Apply the product rule to $- v$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $15$ by $1$ .

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

Simplify.

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

Apply the product rule to $- v$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $6$ by $- 1$ .

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

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

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

Anything raised to $0$ is $1$ .

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

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

$s^{6} - 6 s^{5} v + 15 s^{4} v^{2} - 20 s^{3} v^{3} + 15 s^{2} v^{4} - 6 s v^{5} + {(- v)}^{6}$

Apply the product rule to $- v$ .

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

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

$s^{6} - 6 s^{5} v + 15 s^{4} v^{2} - 20 s^{3} v^{3} + 15 s^{2} v^{4} - 6 s v^{5} + 1 v^{6}$

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

$s^{6} - 6 s^{5} v + 15 s^{4} v^{2} - 20 s^{3} v^{3} + 15 s^{2} v^{4} - 6 s v^{5} + v^{6}$

Expand Using Pascal's Triangle (s-v)^6

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