Expand Using the Binomial Theorem (3x+1)^4

${(3 x + 1)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(3 x)}^{4 - k} \cdot {(1)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

Multiply $1$ by ${(1)}^{0}$ by adding the exponents.

Move ${(1)}^{0}$ .

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

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

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $0$ and $1$ .

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

Simplify $1^{1} \cdot {(3 x)}^{4}$ .

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

Apply the product rule to $3 x$ .

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

Raise $3$ to the power of $4$ .

$81 x^{4} + 4 \cdot {(3 x)}^{3} \cdot {(1)}^{1} + 6 \cdot {(3 x)}^{2} \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Apply the product rule to $3 x$ .

$81 x^{4} + 4 \cdot (3^{3} x^{3}) \cdot {(1)}^{1} + 6 \cdot {(3 x)}^{2} \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

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

$81 x^{4} + 4 \cdot (27 x^{3}) \cdot {(1)}^{1} + 6 \cdot {(3 x)}^{2} \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Multiply $27$ by $4$ .

$81 x^{4} + 108 x^{3} \cdot {(1)}^{1} + 6 \cdot {(3 x)}^{2} \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Evaluate the exponent.

$81 x^{4} + 108 x^{3} \cdot 1 + 6 \cdot {(3 x)}^{2} \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Multiply $108$ by $1$ .

$81 x^{4} + 108 x^{3} + 6 \cdot {(3 x)}^{2} \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Apply the product rule to $3 x$ .

$81 x^{4} + 108 x^{3} + 6 \cdot (3^{2} x^{2}) \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Raise $3$ to the power of $2$ .

$81 x^{4} + 108 x^{3} + 6 \cdot (9 x^{2}) \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Multiply $9$ by $6$ .

$81 x^{4} + 108 x^{3} + 54 x^{2} \cdot {(1)}^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

One to any power is one.

$81 x^{4} + 108 x^{3} + 54 x^{2} \cdot 1 + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Multiply $54$ by $1$ .

$81 x^{4} + 108 x^{3} + 54 x^{2} + 4 \cdot {(3 x)}^{1} \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Simplify.

$81 x^{4} + 108 x^{3} + 54 x^{2} + 4 \cdot (3 x) \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Multiply $3$ by $4$ .

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x \cdot {(1)}^{3} + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

One to any power is one.

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x \cdot 1 + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Multiply $12$ by $1$ .

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1 \cdot {(3 x)}^{0} \cdot {(1)}^{4}$

Multiply $1$ by ${(1)}^{4}$ by adding the exponents.

Move ${(1)}^{4}$ .

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + {(1)}^{4} \cdot 1 \cdot {(3 x)}^{0}$

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

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

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + {(1)}^{4} \cdot 1^{1} \cdot {(3 x)}^{0}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1^{4 + 1} \cdot {(3 x)}^{0}$

Add $4$ and $1$ .

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1^{5} \cdot {(3 x)}^{0}$

Simplify $1^{5} \cdot {(3 x)}^{0}$ .

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1^{5}$

One to any power is one.

$81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1$

Expand Using the Binomial Theorem (3x+1)^4

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