Calculus Examples

${(n + 1)}^{4} - 1$

Step 1

Rewrite ${(n + 1)}^{4}$ as ${({(n + 1)}^{2})}^{2}$ .

${({(n + 1)}^{2})}^{2} - 1$

Step 2

Rewrite $1$ as $1^{2}$ .

${({(n + 1)}^{2})}^{2} - 1^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(n + 1)}^{2}$ and $b = 1$ .

$({(n + 1)}^{2} + 1) ({(n + 1)}^{2} - 1)$

Step 4

Simplify.

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Step 4.1

Rewrite ${(n + 1)}^{2}$ as $(n + 1) (n + 1)$ .

$((n + 1) (n + 1) + 1) ({(n + 1)}^{2} - 1)$

Step 4.2

Expand $(n + 1) (n + 1)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$(n (n + 1) + 1 (n + 1) + 1) ({(n + 1)}^{2} - 1)$

Step 4.2.2

Apply the distributive property.

$(n \cdot n + n \cdot 1 + 1 (n + 1) + 1) ({(n + 1)}^{2} - 1)$

Step 4.2.3

Apply the distributive property.

$(n \cdot n + n \cdot 1 + 1 n + 1 \cdot 1 + 1) ({(n + 1)}^{2} - 1)$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $n$ by $n$ .

$(n^{2} + n \cdot 1 + 1 n + 1 \cdot 1 + 1) ({(n + 1)}^{2} - 1)$

Step 4.3.1.2

Multiply $n$ by $1$ .

$(n^{2} + n + 1 n + 1 \cdot 1 + 1) ({(n + 1)}^{2} - 1)$

Step 4.3.1.3

Multiply $n$ by $1$ .

$(n^{2} + n + n + 1 \cdot 1 + 1) ({(n + 1)}^{2} - 1)$

Step 4.3.1.4

Multiply $1$ by $1$ .

$(n^{2} + n + n + 1 + 1) ({(n + 1)}^{2} - 1)$

Step 4.3.2

Add $n$ and $n$ .

$(n^{2} + 2 n + 1 + 1) ({(n + 1)}^{2} - 1)$

Step 4.4

Add $1$ and $1$ .

$(n^{2} + 2 n + 2) ({(n + 1)}^{2} - 1)$

Step 4.5

Rewrite $1$ as $1^{2}$ .

$(n^{2} + 2 n + 2) ({(n + 1)}^{2} - 1^{2})$

Step 4.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = n + 1$ and $b = 1$ .

$(n^{2} + 2 n + 2) ((n + 1 + 1) (n + 1 - 1))$

Step 4.7

Factor.

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Step 4.7.1

Simplify.

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Step 4.7.1.1

Add $1$ and $1$ .

$(n^{2} + 2 n + 2) ((n + 2) (n + 1 - 1))$

Step 4.7.1.2

Subtract $1$ from $1$ .

$(n^{2} + 2 n + 2) ((n + 2) (n + 0))$

Step 4.7.1.3

Add $n$ and $0$ .

$(n^{2} + 2 n + 2) ((n + 2) n)$

Step 4.7.2

Remove unnecessary parentheses.

$(n^{2} + 2 n + 2) (n + 2) n$