Expand Using the Binomial Theorem (z-k)^6

${(z - k)}^{6}$

Step 1

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

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

Step 2

Expand the summation.

$\frac{6!}{(6 - 0)! 0!} {(z)}^{6 - 0} \cdot {(- k)}^{0} + \frac{6!}{(6 - 1)! 1!} {(z)}^{6 - 1} \cdot {(- k)}^{1} + \dots + \frac{6!}{(6 - 6)! 6!} {(z)}^{6 - 6} \cdot {(- k)}^{6}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(z)}^{6} \cdot {(- k)}^{0} + 6 \cdot {(z)}^{5} \cdot {(- k)}^{1} + \dots + 1 \cdot {(z)}^{0} \cdot {(- k)}^{6}$

Step 4

Simplify the polynomial result.

$z^{6} - 6 z^{5} k + 15 z^{4} k^{2} - 20 z^{3} k^{3} + 15 z^{2} k^{4} - 6 z k^{5} + k^{6}$

Expand Using the Binomial Theorem (z-k)^6

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