Algebra Examples

${(x + 5)}^{3}$

Step 1

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

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(x)}^{3 - k} \cdot {(5)}^{k}$

Step 2

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} {(x)}^{3 - 0} \cdot {(5)}^{0} + \frac{3!}{(3 - 1)! 1!} {(x)}^{3 - 1} \cdot {(5)}^{1} + \frac{3!}{(3 - 2)! 2!} {(x)}^{3 - 2} \cdot {(5)}^{2} + \frac{3!}{(3 - 3)! 3!} {(x)}^{3 - 3} \cdot {(5)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{3} \cdot {(5)}^{0} + 3 \cdot {(x)}^{2} \cdot {(5)}^{1} + 3 \cdot {(x)}^{1} \cdot {(5)}^{2} + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x)}^{3}$ by $1$ .

${(x)}^{3} \cdot {(5)}^{0} + 3 \cdot {(x)}^{2} \cdot {(5)}^{1} + 3 \cdot {(x)}^{1} \cdot {(5)}^{2} + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.2

Anything raised to $0$ is $1$ .

$x^{3} \cdot 1 + 3 \cdot {(x)}^{2} \cdot {(5)}^{1} + 3 \cdot {(x)}^{1} \cdot {(5)}^{2} + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.3

Multiply $x^{3}$ by $1$ .

$x^{3} + 3 \cdot {(x)}^{2} \cdot {(5)}^{1} + 3 \cdot {(x)}^{1} \cdot {(5)}^{2} + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.4

Evaluate the exponent.

$x^{3} + 3 x^{2} \cdot 5 + 3 \cdot {(x)}^{1} \cdot {(5)}^{2} + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.5

Multiply $5$ by $3$ .

$x^{3} + 15 x^{2} + 3 \cdot {(x)}^{1} \cdot {(5)}^{2} + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.6

Simplify.

$x^{3} + 15 x^{2} + 3 \cdot x \cdot {(5)}^{2} + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.7

Raise $5$ to the power of $2$ .

$x^{3} + 15 x^{2} + 3 x \cdot 25 + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.8

Multiply $25$ by $3$ .

$x^{3} + 15 x^{2} + 75 x + 1 \cdot {(x)}^{0} \cdot {(5)}^{3}$

Step 4.9

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

$x^{3} + 15 x^{2} + 75 x + {(x)}^{0} \cdot {(5)}^{3}$

Step 4.10

Anything raised to $0$ is $1$ .

$x^{3} + 15 x^{2} + 75 x + 1 \cdot {(5)}^{3}$

Step 4.11

Multiply ${(5)}^{3}$ by $1$ .

$x^{3} + 15 x^{2} + 75 x + {(5)}^{3}$

Step 4.12

Raise $5$ to the power of $3$ .

$x^{3} + 15 x^{2} + 75 x + 125$