Expand Using the Binomial Theorem (d-5y)^6

${(d - 5 y)}^{6}$

Step 1

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

$\sum_{k = 0}^{6} \frac{6!}{(6 - k)! k!} \cdot {(d)}^{6 - k} \cdot {(- 5 y)}^{k}$

Step 2

Expand the summation.

$\frac{6!}{(6 - 0)! 0!} {(d)}^{6 - 0} \cdot {(- 5 y)}^{0} + \frac{6!}{(6 - 1)! 1!} {(d)}^{6 - 1} \cdot {(- 5 y)}^{1} + \dots + \frac{6!}{(6 - 6)! 6!} {(d)}^{6 - 6} \cdot {(- 5 y)}^{6}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(d)}^{6} \cdot {(- 5 y)}^{0} + 6 \cdot {(d)}^{5} \cdot {(- 5 y)}^{1} + \dots + 1 \cdot {(d)}^{0} \cdot {(- 5 y)}^{6}$

Step 4

Simplify the polynomial result.

$d^{6} - 30 d^{5} y + 375 d^{4} y^{2} - 2500 d^{3} y^{3} + 9375 d^{2} y^{4} - 18750 d y^{5} + 15625 y^{6}$

Expand Using the Binomial Theorem (d-5y)^6

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