Expand Using the Binomial Theorem (u+v)^3

${(u + v)}^{3}$

Step 1

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

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(u)}^{3 - k} \cdot {(v)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

Multiply ${(u)}^{3}$ by $1$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $u^{3}$ by $1$ .

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

Simplify.

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

Simplify.

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

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

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

Anything raised to $0$ is $1$ .

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

Multiply ${(v)}^{3}$ by $1$ .

$u^{3} + 3 u^{2} v + 3 u v^{2} + v^{3}$

Expand Using the Binomial Theorem (u+v)^3

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