Algebra Examples

$(16 n^{3} + 32 n^{2} + 16 n) \div 8 n^{2}$

Step 1

Rewrite the division as a fraction.

$\frac{16 n^{3} + 32 n^{2} + 16 n}{8 n^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $16 n$ out of $16 n^{3} + 32 n^{2} + 16 n$ .

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Step 2.1.1

Factor $16 n$ out of $16 n^{3}$ .

$\frac{16 n (n^{2}) + 32 n^{2} + 16 n}{8 n^{2}}$

Step 2.1.2

Factor $16 n$ out of $32 n^{2}$ .

$\frac{16 n (n^{2}) + 16 n (2 n) + 16 n}{8 n^{2}}$

Step 2.1.3

Factor $16 n$ out of $16 n$ .

$\frac{16 n (n^{2}) + 16 n (2 n) + 16 n (1)}{8 n^{2}}$

Step 2.1.4

Factor $16 n$ out of $16 n (n^{2}) + 16 n (2 n)$ .

$\frac{16 n (n^{2} + 2 n) + 16 n (1)}{8 n^{2}}$

Step 2.1.5

Factor $16 n$ out of $16 n (n^{2} + 2 n) + 16 n (1)$ .

$\frac{16 n (n^{2} + 2 n + 1)}{8 n^{2}}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

$\frac{16 n (n^{2} + 2 n + 1^{2})}{8 n^{2}}$

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 n = 2 \cdot n \cdot 1$

Step 2.2.3

Rewrite the polynomial.

$\frac{16 n (n^{2} + 2 \cdot n \cdot 1 + 1^{2})}{8 n^{2}}$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = n$ and $b = 1$ .

$\frac{16 n {(n + 1)}^{2}}{8 n^{2}}$

Step 3

Cancel the common factor of $16$ and $8$ .

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Step 3.1

Factor $8$ out of $16 n {(n + 1)}^{2}$ .

$\frac{8 (2 n {(n + 1)}^{2})}{8 n^{2}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $8$ out of $8 n^{2}$ .

$\frac{8 (2 n {(n + 1)}^{2})}{8 (n^{2})}$

Step 3.2.2

Cancel the common factor.

$\frac{8 (2 n {(n + 1)}^{2})}{8 n^{2}}$

Step 3.2.3

Rewrite the expression.

$\frac{2 n {(n + 1)}^{2}}{n^{2}}$

Step 4

Cancel the common factor of $n$ and $n^{2}$ .

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Step 4.1

Factor $n$ out of $2 n {(n + 1)}^{2}$ .

$\frac{n (2 {(n + 1)}^{2})}{n^{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $n$ out of $n^{2}$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\frac{2 {(n + 1)}^{2}}{n}$

Step 5

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

$\frac{2 ((n + 1) (n + 1))}{n}$

Step 6

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

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Step 6.1

Apply the distributive property.

$\frac{2 (n (n + 1) + 1 (n + 1))}{n}$

Step 6.2

Apply the distributive property.

$\frac{2 (n \cdot n + n \cdot 1 + 1 (n + 1))}{n}$

Step 6.3

Apply the distributive property.

$\frac{2 (n \cdot n + n \cdot 1 + 1 n + 1 \cdot 1)}{n}$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $n$ by $n$ .

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

Step 7.1.2

Multiply $n$ by $1$ .

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

Step 7.1.3

Multiply $n$ by $1$ .

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

Step 7.1.4

Multiply $1$ by $1$ .

$\frac{2 (n^{2} + n + n + 1)}{n}$

Step 7.2

Add $n$ and $n$ .

$\frac{2 (n^{2} + 2 n + 1)}{n}$

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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Step 9.1

Multiply $2$ by $2$ .

$\frac{2 n^{2} + 4 n + 2 \cdot 1}{n}$

Step 9.2

Multiply $2$ by $1$ .

$\frac{2 n^{2} + 4 n + 2}{n}$

Step 10

Split the fraction $\frac{2 n^{2} + 4 n + 2}{n}$ into two fractions.

$\frac{2 n^{2} + 4 n}{n} + \frac{2}{n}$

Step 11

Split the fraction $\frac{2 n^{2} + 4 n}{n}$ into two fractions.

$\frac{2 n^{2}}{n} + \frac{4 n}{n} + \frac{2}{n}$

Step 12

Cancel the common factor of $n^{2}$ and $n$ .

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Step 12.1

Factor $n$ out of $2 n^{2}$ .

$\frac{n (2 n)}{n} + \frac{4 n}{n} + \frac{2}{n}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

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

$\frac{n (2 n)}{n^{1}} + \frac{4 n}{n} + \frac{2}{n}$

Step 12.2.2

Factor $n$ out of $n^{1}$ .

$\frac{n (2 n)}{n \cdot 1} + \frac{4 n}{n} + \frac{2}{n}$

Step 12.2.3

Cancel the common factor.

$\frac{n (2 n)}{n \cdot 1} + \frac{4 n}{n} + \frac{2}{n}$

Step 12.2.4

Rewrite the expression.

$\frac{2 n}{1} + \frac{4 n}{n} + \frac{2}{n}$

Step 12.2.5

Divide $2 n$ by $1$ .

$2 n + \frac{4 n}{n} + \frac{2}{n}$

Step 13

Cancel the common factor of $n$ .

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Step 13.1

Cancel the common factor.

$2 n + \frac{4 n}{n} + \frac{2}{n}$

Step 13.2

Divide $4$ by $1$ .

$2 n + 4 + \frac{2}{n}$