Expand Using the Binomial Theorem (q-r)^7

${(q - r)}^{7}$

Step 1

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

$\sum_{k = 0}^{7} \frac{7!}{(7 - k)! k!} \cdot {(q)}^{7 - k} \cdot {(- r)}^{k}$

Step 2

Expand the summation.

$\frac{7!}{(7 - 0)! 0!} {(q)}^{7 - 0} \cdot {(- r)}^{0} + \frac{7!}{(7 - 1)! 1!} {(q)}^{7 - 1} \cdot {(- r)}^{1} + \dots + \frac{7!}{(7 - 7)! 7!} {(q)}^{7 - 7} \cdot {(- r)}^{7}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(q)}^{7} \cdot {(- r)}^{0} + 7 \cdot {(q)}^{6} \cdot {(- r)}^{1} + \dots + 1 \cdot {(q)}^{0} \cdot {(- r)}^{7}$

Step 4

Simplify the polynomial result.

$q^{7} - 7 q^{6} r + 21 q^{5} r^{2} - 35 q^{4} r^{3} + 35 q^{3} r^{4} - 21 q^{2} r^{5} + 7 q r^{6} - r^{7}$

Expand Using the Binomial Theorem (q-r)^7

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