Expand Using the Binomial Theorem (u-v)^4

${(u - v)}^{4}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(u)}^{4 - k} \cdot {(- v)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(u)}^{4} \cdot {(- v)}^{0} + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Step 4

Simplify each term.

Multiply ${(u)}^{4}$ by $1$ .

${(u)}^{4} \cdot {(- v)}^{0} + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Apply the product rule to $- v$ .

$u^{4} \cdot ({(- 1)}^{0} v^{0}) + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} u^{4} v^{0} + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Anything raised to $0$ is $1$ .

$1 u^{4} v^{0} + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Multiply $u^{4}$ by $1$ .

$u^{4} v^{0} + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Anything raised to $0$ is $1$ .

$u^{4} \cdot 1 + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Multiply $u^{4}$ by $1$ .

$u^{4} + 4 \cdot {(u)}^{3} \cdot {(- v)}^{1} + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Simplify.

$u^{4} + 4 u^{3} \cdot (- v) + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Rewrite using the commutative property of multiplication.

$u^{4} + 4 \cdot - 1 u^{3} v + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Multiply $4$ by $- 1$ .

$u^{4} - 4 u^{3} v + 6 \cdot {(u)}^{2} \cdot {(- v)}^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Apply the product rule to $- v$ .

$u^{4} - 4 u^{3} v + 6 u^{2} \cdot ({(- 1)}^{2} v^{2}) + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Rewrite using the commutative property of multiplication.

$u^{4} - 4 u^{3} v + 6 \cdot {(- 1)}^{2} u^{2} v^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

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

$u^{4} - 4 u^{3} v + 6 \cdot 1 u^{2} v^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Multiply $6$ by $1$ .

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} + 4 \cdot {(u)}^{1} \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Simplify.

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} + 4 \cdot u \cdot {(- v)}^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Apply the product rule to $- v$ .

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} + 4 u \cdot ({(- 1)}^{3} v^{3}) + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Rewrite using the commutative property of multiplication.

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} + 4 \cdot {(- 1)}^{3} u v^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

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

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} + 4 \cdot - 1 u v^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

Multiply $4$ by $- 1$ .

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} - 4 u v^{3} + 1 \cdot {(u)}^{0} \cdot {(- v)}^{4}$

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

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} - 4 u v^{3} + {(u)}^{0} \cdot {(- v)}^{4}$

Anything raised to $0$ is $1$ .

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} - 4 u v^{3} + 1 \cdot {(- v)}^{4}$

Multiply ${(- v)}^{4}$ by $1$ .

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} - 4 u v^{3} + {(- v)}^{4}$

Apply the product rule to $- v$ .

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} - 4 u v^{3} + {(- 1)}^{4} v^{4}$

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

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} - 4 u v^{3} + 1 v^{4}$

Multiply $v^{4}$ by $1$ .

$u^{4} - 4 u^{3} v + 6 u^{2} v^{2} - 4 u v^{3} + v^{4}$

Expand Using the Binomial Theorem (u-v)^4

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