Expand Using Pascal's Triangle (u-v)^5

${(u - v)}^{5}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

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 ${(u - v)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

Substitute the actual values of $a$ $u$ and $b$ $- v$ into the expression.

$1 {(u)}^{5} {(- v)}^{0} + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Step 4

Simplify each term.

Multiply ${(u)}^{5}$ by $1$ .

${(u)}^{5} {(- v)}^{0} + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Apply the product rule to $- v$ .

$u^{5} ({(- 1)}^{0} v^{0}) + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} u^{5} v^{0} + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Anything raised to $0$ is $1$ .

$1 u^{5} v^{0} + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Multiply $u^{5}$ by $1$ .

$u^{5} v^{0} + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Anything raised to $0$ is $1$ .

$u^{5} \cdot 1 + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Multiply $u^{5}$ by $1$ .

$u^{5} + 5 {(u)}^{4} {(- v)}^{1} + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Simplify.

$u^{5} + 5 u^{4} (- v) + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Rewrite using the commutative property of multiplication.

$u^{5} + 5 \cdot - 1 u^{4} v + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Multiply $5$ by $- 1$ .

$u^{5} - 5 u^{4} v + 10 {(u)}^{3} {(- v)}^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Apply the product rule to $- v$ .

$u^{5} - 5 u^{4} v + 10 u^{3} ({(- 1)}^{2} v^{2}) + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Rewrite using the commutative property of multiplication.

$u^{5} - 5 u^{4} v + 10 \cdot {(- 1)}^{2} u^{3} v^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

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

$u^{5} - 5 u^{4} v + 10 \cdot 1 u^{3} v^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Multiply $10$ by $1$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} + 10 {(u)}^{2} {(- v)}^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Apply the product rule to $- v$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} + 10 u^{2} ({(- 1)}^{3} v^{3}) + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Rewrite using the commutative property of multiplication.

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} + 10 \cdot {(- 1)}^{3} u^{2} v^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

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

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} + 10 \cdot - 1 u^{2} v^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Multiply $10$ by $- 1$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 {(u)}^{1} {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Simplify.

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u {(- v)}^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Apply the product rule to $- v$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u ({(- 1)}^{4} v^{4}) + 1 {(u)}^{0} {(- v)}^{5}$

Rewrite using the commutative property of multiplication.

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 \cdot {(- 1)}^{4} u v^{4} + 1 {(u)}^{0} {(- v)}^{5}$

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

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 \cdot 1 u v^{4} + 1 {(u)}^{0} {(- v)}^{5}$

Multiply $5$ by $1$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u v^{4} + 1 {(u)}^{0} {(- v)}^{5}$

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

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u v^{4} + {(u)}^{0} {(- v)}^{5}$

Anything raised to $0$ is $1$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u v^{4} + 1 {(- v)}^{5}$

Multiply ${(- v)}^{5}$ by $1$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u v^{4} + {(- v)}^{5}$

Apply the product rule to $- v$ .

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u v^{4} + {(- 1)}^{5} v^{5}$

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

$u^{5} - 5 u^{4} v + 10 u^{3} v^{2} - 10 u^{2} v^{3} + 5 u v^{4} - v^{5}$

Expand Using Pascal's Triangle (u-v)^5

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