Expand Using the Binomial Theorem (q+r)^4

${(q + r)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(q)}^{4 - k} \cdot {(r)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} {(q)}^{4 - 0} \cdot {(r)}^{0} + \frac{4!}{(4 - 1)! 1!} {(q)}^{4 - 1} \cdot {(r)}^{1} + \frac{4!}{(4 - 2)! 2!} {(q)}^{4 - 2} \cdot {(r)}^{2} + \frac{4!}{(4 - 3)! 3!} {(q)}^{4 - 3} \cdot {(r)}^{3} + \frac{4!}{(4 - 4)! 4!} {(q)}^{4 - 4} \cdot {(r)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(q)}^{4} \cdot {(r)}^{0} + 4 \cdot {(q)}^{3} \cdot {(r)}^{1} + 6 \cdot {(q)}^{2} \cdot {(r)}^{2} + 4 \cdot {(q)}^{1} \cdot {(r)}^{3} + 1 \cdot {(q)}^{0} \cdot {(r)}^{4}$

Step 4

Simplify each term.

Multiply ${(q)}^{4}$ by $1$ .

${(q)}^{4} \cdot {(r)}^{0} + 4 \cdot {(q)}^{3} \cdot {(r)}^{1} + 6 \cdot {(q)}^{2} \cdot {(r)}^{2} + 4 \cdot {(q)}^{1} \cdot {(r)}^{3} + 1 \cdot {(q)}^{0} \cdot {(r)}^{4}$

Anything raised to $0$ is $1$ .

$q^{4} \cdot 1 + 4 \cdot {(q)}^{3} \cdot {(r)}^{1} + 6 \cdot {(q)}^{2} \cdot {(r)}^{2} + 4 \cdot {(q)}^{1} \cdot {(r)}^{3} + 1 \cdot {(q)}^{0} \cdot {(r)}^{4}$

Multiply $q^{4}$ by $1$ .

$q^{4} + 4 \cdot {(q)}^{3} \cdot {(r)}^{1} + 6 \cdot {(q)}^{2} \cdot {(r)}^{2} + 4 \cdot {(q)}^{1} \cdot {(r)}^{3} + 1 \cdot {(q)}^{0} \cdot {(r)}^{4}$

Simplify.

$q^{4} + 4 q^{3} \cdot r + 6 \cdot {(q)}^{2} \cdot {(r)}^{2} + 4 \cdot {(q)}^{1} \cdot {(r)}^{3} + 1 \cdot {(q)}^{0} \cdot {(r)}^{4}$

Simplify.

$q^{4} + 4 q^{3} r + 6 q^{2} r^{2} + 4 \cdot q \cdot {(r)}^{3} + 1 \cdot {(q)}^{0} \cdot {(r)}^{4}$

Multiply ${(q)}^{0}$ by $1$ .

$q^{4} + 4 q^{3} r + 6 q^{2} r^{2} + 4 q r^{3} + {(q)}^{0} \cdot {(r)}^{4}$

Anything raised to $0$ is $1$ .

$q^{4} + 4 q^{3} r + 6 q^{2} r^{2} + 4 q r^{3} + 1 \cdot {(r)}^{4}$

Multiply ${(r)}^{4}$ by $1$ .

$q^{4} + 4 q^{3} r + 6 q^{2} r^{2} + 4 q r^{3} + r^{4}$

Expand Using the Binomial Theorem (q+r)^4

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