Pre-Algebra Examples

$\frac{21 x^{2} + x - 2}{3 x - 3} \div \frac{49 x^{2} - 4}{x - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{21 x^{2} + x - 2}{3 x - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 2

Factor by grouping.

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Step 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 $a \cdot c = 21 \cdot - 2 = - 42$ and whose sum is $b = 1$ .

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Step 2.1.1

Multiply by $1$ .

$\frac{21 x^{2} + 1 x - 2}{3 x - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 2.1.2

Rewrite $1$ as $- 6$ plus $7$

$\frac{21 x^{2} + (- 6 + 7) x - 2}{3 x - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 2.1.3

Apply the distributive property.

$\frac{21 x^{2} - 6 x + 7 x - 2}{3 x - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(21 x^{2} - 6 x) + 7 x - 2}{3 x - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 2.2.2

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

$\frac{3 x (7 x - 2) + 1 (7 x - 2)}{3 x - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $7 x - 2$ .

$\frac{(7 x - 2) (3 x + 1)}{3 x - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 3

Simplify terms.

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Step 3.1

Factor $3$ out of $3 x - 3$ .

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Step 3.1.1

Factor $3$ out of $3 x$ .

$\frac{(7 x - 2) (3 x + 1)}{3 (x) - 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 3.1.2

Factor $3$ out of $- 3$ .

$\frac{(7 x - 2) (3 x + 1)}{3 (x) + 3 (- 1)} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 3.1.3

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

$\frac{(7 x - 2) (3 x + 1)}{3 (x - 1)} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 3.2

Cancel the common factor of $x - 1$ .

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Step 3.2.1

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

$\frac{(7 x - 2) (3 x + 1)}{(x - 1) \cdot 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 3.2.2

Cancel the common factor.

$\frac{(7 x - 2) (3 x + 1)}{(x - 1) \cdot 3} \cdot \frac{x - 1}{49 x^{2} - 4}$

Step 3.2.3

Rewrite the expression.

$\frac{(7 x - 2) (3 x + 1)}{3} \cdot \frac{1}{49 x^{2} - 4}$

Step 3.3

Multiply $\frac{(7 x - 2) (3 x + 1)}{3}$ by $\frac{1}{49 x^{2} - 4}$ .

$\frac{(7 x - 2) (3 x + 1)}{3 (49 x^{2} - 4)}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

$\frac{(7 x - 2) (3 x + 1)}{3 ({(7 x)}^{2} - 4)}$

Step 4.2

Rewrite $4$ as $2^{2}$ .

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7 x$ and $b = 2$ .

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

Step 5

Cancel the common factor of $7 x - 2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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