Pre-Algebra Examples

$\frac{12 x^{2} - 14 x + 4}{3 x} \div \frac{3 x^{2} + x - 2}{2 x^{2} + 2 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{12 x^{2} - 14 x + 4}{3 x} \cdot \frac{2 x^{2} + 2 x}{3 x^{2} + x - 2}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $12 x^{2} - 14 x + 4$ .

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Step 2.1.1

Factor $2$ out of $12 x^{2}$ .

$\frac{2 (6 x^{2}) - 14 x + 4}{3 x} \cdot \frac{2 x^{2} + 2 x}{3 x^{2} + x - 2}$

Step 2.1.2

Factor $2$ out of $- 14 x$ .

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

Step 2.1.3

Factor $2$ out of $4$ .

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

Step 2.1.4

Factor $2$ out of $2 (6 x^{2}) + 2 (- 7 x)$ .

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

Step 2.1.5

Factor $2$ out of $2 (6 x^{2} - 7 x) + 2 (2)$ .

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

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = 6 \cdot 2 = 12$ and whose sum is $b = - 7$ .

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Step 2.2.1.1

Factor $- 7$ out of $- 7 x$ .

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

Step 2.2.1.2

Rewrite $- 7$ as $- 3$ plus $- 4$

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.2.2

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

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

Step 2.2.3

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Factor $2 x$ out of $2 x$ .

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

Step 3.3

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

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

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

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Step 4.1.1

Multiply by $1$ .

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

Step 4.1.2

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

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2

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

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

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 5

Cancel the common factor of $3 x - 2$ .

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Step 5.1

Factor $3 x - 2$ out of $2 (2 x - 1) (3 x - 2)$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $x$ .

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Step 6.1

Factor $x$ out of $3 x$ .

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

Step 6.2

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

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Multiply $\frac{2 (2 x - 1)}{3}$ by $\frac{2 (x + 1)}{x + 1}$ .

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

Step 8

Multiply $2$ by $2$ .

$\frac{4 (2 x - 1) (x + 1)}{3 (x + 1)}$

Step 9

Cancel the common factor of $x + 1$ .

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Step 9.1

Cancel the common factor.

$\frac{4 (2 x - 1) (x + 1)}{3 (x + 1)}$

Step 9.2

Rewrite the expression.

$\frac{4 (2 x - 1)}{3}$

Step 10

Apply the distributive property.

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

Step 11

Multiply $2$ by $4$ .

$\frac{8 x + 4 \cdot - 1}{3}$

Step 12

Multiply $4$ by $- 1$ .

$\frac{8 x - 4}{3}$

Step 13

Split the fraction $\frac{8 x - 4}{3}$ into two fractions.

$\frac{8 x}{3} + \frac{- 4}{3}$

Step 14

Move the negative in front of the fraction.

$\frac{8 x}{3} - \frac{4}{3}$