Expand Using the Binomial Theorem (ax+by)^5

${(a x + b y)}^{5}$

Step 1

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

$\sum_{k = 0}^{5} \frac{5!}{(5 - k)! k!} \cdot {(a x)}^{5 - k} \cdot {(b y)}^{k}$

Step 2

Expand the summation.

$\frac{5!}{(5 - 0)! 0!} \cdot {(a x)}^{5 - 0} \cdot {(b y)}^{0} + \frac{5!}{(5 - 1)! 1!} \cdot {(a x)}^{5 - 1} \cdot {(b y)}^{1} + \dots + \frac{5!}{(5 - 5)! 5!} \cdot {(a x)}^{5 - 5} \cdot {(b y)}^{5}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(a x)}^{5} \cdot {(b y)}^{0} + 5 \cdot {(a x)}^{4} \cdot {(b y)}^{1} + \dots + 1 \cdot {(a x)}^{0} \cdot {(b y)}^{5}$

Step 4

Simplify the polynomial result.

$a^{5} x^{5} + 5 a^{4} x^{4} b y + 10 a^{3} x^{3} b^{2} y^{2} + 10 a^{2} x^{2} b^{3} y^{3} + 5 a x b^{4} y^{4} + b^{5} y^{5}$

Expand Using the Binomial Theorem (ax+by)^5

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