Expand Using the Binomial Theorem (2x^3y+y)^7

${(2 x^{3} y + y)}^{7}$

Step 1

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

$\sum_{k = 0}^{7} \frac{7!}{(7 - k)! k!} \cdot {(2 x^{3} y)}^{7 - k} \cdot {(y)}^{k}$

Step 2

Expand the summation.

$\frac{7!}{(7 - 0)! 0!} \cdot {(2 x^{3} y)}^{7 - 0} \cdot {(y)}^{0} + \frac{7!}{(7 - 1)! 1!} \cdot {(2 x^{3} y)}^{7 - 1} \cdot {(y)}^{1} + \dots + \frac{7!}{(7 - 7)! 7!} \cdot {(2 x^{3} y)}^{7 - 7} \cdot {(y)}^{7}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(2 x^{3} y)}^{7} \cdot {(y)}^{0} + 7 \cdot {(2 x^{3} y)}^{6} \cdot {(y)}^{1} + \dots + 1 \cdot {(2 x^{3} y)}^{0} \cdot {(y)}^{7}$

Step 4

Simplify the polynomial result.

$128 x^{21} y^{7} + 448 x^{18} y^{7} + 672 x^{15} y^{7} + 560 x^{12} y^{7} + 280 x^{9} y^{7} + 84 x^{6} y^{7} + 14 x^{3} y^{7} + y^{7}$

Expand Using the Binomial Theorem (2x^3y+y)^7

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