Expand Using the Binomial Theorem (X+Y)^4

${(X + Y)}^{4}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(X)}^{4} \cdot {(Y)}^{0} + 4 \cdot {(X)}^{3} \cdot {(Y)}^{1} + 6 \cdot {(X)}^{2} \cdot {(Y)}^{2} + 4 \cdot {(X)}^{1} \cdot {(Y)}^{3} + 1 \cdot {(X)}^{0} \cdot {(Y)}^{4}$

Step 4

Simplify each term.

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

${(X)}^{4} \cdot {(Y)}^{0} + 4 \cdot {(X)}^{3} \cdot {(Y)}^{1} + 6 \cdot {(X)}^{2} \cdot {(Y)}^{2} + 4 \cdot {(X)}^{1} \cdot {(Y)}^{3} + 1 \cdot {(X)}^{0} \cdot {(Y)}^{4}$

Anything raised to $0$ is $1$ .

$X^{4} \cdot 1 + 4 \cdot {(X)}^{3} \cdot {(Y)}^{1} + 6 \cdot {(X)}^{2} \cdot {(Y)}^{2} + 4 \cdot {(X)}^{1} \cdot {(Y)}^{3} + 1 \cdot {(X)}^{0} \cdot {(Y)}^{4}$

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

$X^{4} + 4 \cdot {(X)}^{3} \cdot {(Y)}^{1} + 6 \cdot {(X)}^{2} \cdot {(Y)}^{2} + 4 \cdot {(X)}^{1} \cdot {(Y)}^{3} + 1 \cdot {(X)}^{0} \cdot {(Y)}^{4}$

Simplify.

$X^{4} + 4 X^{3} \cdot Y + 6 \cdot {(X)}^{2} \cdot {(Y)}^{2} + 4 \cdot {(X)}^{1} \cdot {(Y)}^{3} + 1 \cdot {(X)}^{0} \cdot {(Y)}^{4}$

Simplify.

$X^{4} + 4 X^{3} Y + 6 X^{2} Y^{2} + 4 \cdot X \cdot {(Y)}^{3} + 1 \cdot {(X)}^{0} \cdot {(Y)}^{4}$

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

$X^{4} + 4 X^{3} Y + 6 X^{2} Y^{2} + 4 X Y^{3} + {(X)}^{0} \cdot {(Y)}^{4}$

Anything raised to $0$ is $1$ .

$X^{4} + 4 X^{3} Y + 6 X^{2} Y^{2} + 4 X Y^{3} + 1 \cdot {(Y)}^{4}$

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

$X^{4} + 4 X^{3} Y + 6 X^{2} Y^{2} + 4 X Y^{3} + Y^{4}$

Expand Using the Binomial Theorem (X+Y)^4

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