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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $- w$ .

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

Rewrite using the commutative property of multiplication.

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

Anything raised to $0$ is $1$ .

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

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

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

Anything raised to $0$ is $1$ .

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

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

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Multiply $4$ by $- 1$ .

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

Apply the product rule to $- w$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $6$ by $1$ .

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

Simplify.

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

Apply the product rule to $- w$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

Multiply $4$ by $- 1$ .

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

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

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

Anything raised to $0$ is $1$ .

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

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

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

Apply the product rule to $- w$ .

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

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

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

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

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

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

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