Expand Using the Binomial Theorem (X+Y)^8

${(X + Y)}^{8}$

Step 1

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

$\sum_{k = 0}^{8} \frac{8!}{(8 - k)! k!} \cdot {(X)}^{8 - k} \cdot {(Y)}^{k}$

Step 2

Expand the summation.

$\frac{8!}{(8 - 0)! 0!} {(X)}^{8 - 0} \cdot {(Y)}^{0} + \frac{8!}{(8 - 1)! 1!} {(X)}^{8 - 1} \cdot {(Y)}^{1} + \dots + \frac{8!}{(8 - 8)! 8!} {(X)}^{8 - 8} \cdot {(Y)}^{8}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(X)}^{8} \cdot {(Y)}^{0} + 8 \cdot {(X)}^{7} \cdot {(Y)}^{1} + \dots + 1 \cdot {(X)}^{0} \cdot {(Y)}^{8}$

Step 4

Simplify the polynomial result.

$X^{8} + 8 X^{7} Y + 28 X^{6} Y^{2} + 56 X^{5} Y^{3} + 70 X^{4} Y^{4} + 56 X^{3} Y^{5} + 28 X^{2} Y^{6} + 8 X Y^{7} + Y^{8}$

Expand Using the Binomial Theorem (X+Y)^8

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