Algebra Examples

$\frac{12 x + 6}{21 x^{2} - 21} \div \frac{6 x^{2} + 9 x + 3}{7 x^{3} - 7 x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{12 x + 6}{21 x^{2} - 21} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $12 x + 6$ and $21 x^{2} - 21$ .

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Step 2.1.1

Factor $3$ out of $12 x$ .

$\frac{3 (4 x) + 6}{21 x^{2} - 21} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.1.2

Factor $3$ out of $6$ .

$\frac{3 (4 x) + 3 (2)}{21 x^{2} - 21} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.1.3

Factor $3$ out of $3 (4 x) + 3 (2)$ .

$\frac{3 (4 x + 2)}{21 x^{2} - 21} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.1.4

Cancel the common factors.

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Step 2.1.4.1

Factor $3$ out of $21 x^{2}$ .

$\frac{3 (4 x + 2)}{3 (7 x^{2}) - 21} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.1.4.2

Factor $3$ out of $- 21$ .

$\frac{3 (4 x + 2)}{3 (7 x^{2}) + 3 (- 7)} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.1.4.3

Factor $3$ out of $3 (7 x^{2}) + 3 (- 7)$ .

$\frac{3 (4 x + 2)}{3 (7 x^{2} - 7)} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.1.4.4

Cancel the common factor.

$\frac{3 (4 x + 2)}{3 (7 x^{2} - 7)} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.1.4.5

Rewrite the expression.

$\frac{4 x + 2}{7 x^{2} - 7} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.2

Factor $2$ out of $4 x + 2$ .

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Step 2.2.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 2}{7 x^{2} - 7} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.2.2

Factor $2$ out of $2$ .

$\frac{2 (2 x) + 2 (1)}{7 x^{2} - 7} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 2.2.3

Factor $2$ out of $2 (2 x) + 2 (1)$ .

$\frac{2 (2 x + 1)}{7 x^{2} - 7} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $7$ out of $7 x^{2} - 7$ .

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Step 3.1.1

Factor $7$ out of $7 x^{2}$ .

$\frac{2 (2 x + 1)}{7 (x^{2}) - 7} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 3.1.2

Factor $7$ out of $- 7$ .

$\frac{2 (2 x + 1)}{7 (x^{2}) + 7 (- 1)} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 3.1.3

Factor $7$ out of $7 (x^{2}) + 7 (- 1)$ .

$\frac{2 (2 x + 1)}{7 (x^{2} - 1)} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 3.2

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

$\frac{2 (2 x + 1)}{7 (x^{2} - 1^{2})} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 3.3

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

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{3} - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 4

Factor $7 x^{2}$ out of $7 x^{3} - 7 x^{2}$ .

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Step 4.1

Factor $7 x^{2}$ out of $7 x^{3}$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x) - 7 x^{2}}{6 x^{2} + 9 x + 3}$

Step 4.2

Factor $7 x^{2}$ out of $- 7 x^{2}$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x) + 7 x^{2} (- 1)}{6 x^{2} + 9 x + 3}$

Step 4.3

Factor $7 x^{2}$ out of $7 x^{2} (x) + 7 x^{2} (- 1)$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{6 x^{2} + 9 x + 3}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $3$ out of $6 x^{2} + 9 x + 3$ .

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Step 5.1.1

Factor $3$ out of $6 x^{2}$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2}) + 9 x + 3}$

Step 5.1.2

Factor $3$ out of $9 x$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2}) + 3 (3 x) + 3}$

Step 5.1.3

Factor $3$ out of $3$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2}) + 3 (3 x) + 3 (1)}$

Step 5.1.4

Factor $3$ out of $3 (2 x^{2}) + 3 (3 x)$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2} + 3 x) + 3 (1)}$

Step 5.1.5

Factor $3$ out of $3 (2 x^{2} + 3 x) + 3 (1)$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2} + 3 x + 1)}$

Step 5.2

Factor by grouping.

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Step 5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = 3$ .

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Step 5.2.1.1

Factor $3$ out of $3 x$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2} + 3 (x) + 1)}$

Step 5.2.1.2

Rewrite $3$ as $1$ plus $2$

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2} + (1 + 2) x + 1)}$

Step 5.2.1.3

Apply the distributive property.

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2} + 1 x + 2 x + 1)}$

Step 5.2.1.4

Multiply $x$ by $1$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x^{2} + x + 2 x + 1)}$

Step 5.2.2

Factor out the greatest common factor from each group.

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Step 5.2.2.1

Group the first two terms and the last two terms.

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 ((2 x^{2} + x) + 2 x + 1)}$

Step 5.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (x (2 x + 1) + 1 (2 x + 1))}$

Step 5.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 ((2 x + 1) (x + 1))}$

$\frac{2 (2 x + 1)}{7 (x + 1) (x - 1)} \cdot \frac{7 x^{2} (x - 1)}{3 (2 x + 1) (x + 1)}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{2 (2 x + 1) (7 x^{2} (x - 1))}{7 (x + 1) (x - 1) (3 (2 x + 1) (x + 1))}$

Step 6.2

Cancel the common factor of $2 x + 1$ .

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Step 6.2.1

Cancel the common factor.

$\frac{2 (2 x + 1) (7 x^{2} (x - 1))}{7 (x + 1) (x - 1) (3 (2 x + 1) (x + 1))}$

Step 6.2.2

Rewrite the expression.

$\frac{2 (7 x^{2} (x - 1))}{7 (x + 1) (x - 1) (3 (x + 1))}$

Step 6.3

Cancel the common factor of $7$ .

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Step 6.3.1

Cancel the common factor.

$\frac{2 (7 x^{2} (x - 1))}{7 (x + 1) (x - 1) (3 (x + 1))}$

Step 6.3.2

Rewrite the expression.

$\frac{2 (x^{2} (x - 1))}{((x + 1) (x - 1)) (3 (x + 1))}$

Step 6.4

Cancel the common factor of $x - 1$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 (x^{2} (x - 1))}{(x + 1) (x - 1) (3 (x + 1))}$

Step 6.4.2

Rewrite the expression.

$\frac{2 (x^{2})}{(x + 1) (3 (x + 1))}$

Step 7

Simplify the denominator.

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Step 7.1

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

$\frac{2 x^{2}}{3 ({(x + 1)}^{1} (x + 1))}$

Step 7.2

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

$\frac{2 x^{2}}{3 ({(x + 1)}^{1} {(x + 1)}^{1})}$

Step 7.3

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

$\frac{2 x^{2}}{3 {(x + 1)}^{1 + 1}}$

Step 7.4

Add $1$ and $1$ .

$\frac{2 x^{2}}{3 {(x + 1)}^{2}}$