Expand Using the Binomial Theorem (5x-1)^3

${(5 x - 1)}^{3}$

Step 1

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

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(5 x)}^{3 - k} \cdot {(- 1)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $5 x$ .

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

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

$125 x^{3} \cdot {(- 1)}^{0} + 3 \cdot {(5 x)}^{2} \cdot {(- 1)}^{1} + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Anything raised to $0$ is $1$ .

$125 x^{3} \cdot 1 + 3 \cdot {(5 x)}^{2} \cdot {(- 1)}^{1} + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Multiply $125$ by $1$ .

$125 x^{3} + 3 \cdot {(5 x)}^{2} \cdot {(- 1)}^{1} + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Apply the product rule to $5 x$ .

$125 x^{3} + 3 \cdot (5^{2} x^{2}) \cdot {(- 1)}^{1} + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

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

$125 x^{3} + 3 \cdot (25 x^{2}) \cdot {(- 1)}^{1} + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Multiply $25$ by $3$ .

$125 x^{3} + 75 x^{2} \cdot {(- 1)}^{1} + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Evaluate the exponent.

$125 x^{3} + 75 x^{2} \cdot - 1 + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Multiply $- 1$ by $75$ .

$125 x^{3} - 75 x^{2} + 3 \cdot {(5 x)}^{1} \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Simplify.

$125 x^{3} - 75 x^{2} + 3 \cdot (5 x) \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Multiply $5$ by $3$ .

$125 x^{3} - 75 x^{2} + 15 x \cdot {(- 1)}^{2} + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Raise $- 1$ to the power of $2$ .

$125 x^{3} - 75 x^{2} + 15 x \cdot 1 + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Multiply $15$ by $1$ .

$125 x^{3} - 75 x^{2} + 15 x + 1 \cdot {(5 x)}^{0} \cdot {(- 1)}^{3}$

Multiply ${(5 x)}^{0}$ by $1$ .

$125 x^{3} - 75 x^{2} + 15 x + {(5 x)}^{0} \cdot {(- 1)}^{3}$

Apply the product rule to $5 x$ .

$125 x^{3} - 75 x^{2} + 15 x + 5^{0} x^{0} \cdot {(- 1)}^{3}$

Anything raised to $0$ is $1$ .

$125 x^{3} - 75 x^{2} + 15 x + 1 x^{0} \cdot {(- 1)}^{3}$

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

$125 x^{3} - 75 x^{2} + 15 x + x^{0} \cdot {(- 1)}^{3}$

Anything raised to $0$ is $1$ .

$125 x^{3} - 75 x^{2} + 15 x + 1 \cdot {(- 1)}^{3}$

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

$125 x^{3} - 75 x^{2} + 15 x + {(- 1)}^{3}$

Raise $- 1$ to the power of $3$ .

$125 x^{3} - 75 x^{2} + 15 x - 1$

Expand Using the Binomial Theorem (5x-1)^3

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