Pre-Algebra Examples

$\frac{\frac{2 x^{2} + x - 21}{5 x^{2} + 12 x + 7}}{\frac{3 x - 9}{5 x + 7}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{2} + x - 21}{5 x^{2} + 12 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

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

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Step 2.1.1

Multiply by $1$ .

$\frac{2 x^{2} + 1 x - 21}{5 x^{2} + 12 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 2.1.2

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

$\frac{2 x^{2} + (- 6 + 7) x - 21}{5 x^{2} + 12 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 2.1.3

Apply the distributive property.

$\frac{2 x^{2} - 6 x + 7 x - 21}{5 x^{2} + 12 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

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.

$\frac{(2 x^{2} - 6 x) + 7 x - 21}{5 x^{2} + 12 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 2.2.2

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

$\frac{2 x (x - 3) + 7 (x - 3)}{5 x^{2} + 12 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 2.3

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

$\frac{(x - 3) (2 x + 7)}{5 x^{2} + 12 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 5 \cdot 7 = 35$ and whose sum is $b = 12$ .

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Step 3.1.1

Factor $12$ out of $12 x$ .

$\frac{(x - 3) (2 x + 7)}{5 x^{2} + 12 (x) + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 3.1.2

Rewrite $12$ as $5$ plus $7$

$\frac{(x - 3) (2 x + 7)}{5 x^{2} + (5 + 7) x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 3.1.3

Apply the distributive property.

$\frac{(x - 3) (2 x + 7)}{5 x^{2} + 5 x + 7 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(x - 3) (2 x + 7)}{(5 x^{2} + 5 x) + 7 x + 7} \cdot \frac{5 x + 7}{3 x - 9}$

Step 3.2.2

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

$\frac{(x - 3) (2 x + 7)}{5 x (x + 1) + 7 (x + 1)} \cdot \frac{5 x + 7}{3 x - 9}$

Step 3.3

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

$\frac{(x - 3) (2 x + 7)}{(x + 1) (5 x + 7)} \cdot \frac{5 x + 7}{3 x - 9}$

Step 4

Cancel the common factor of $5 x + 7$ .

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Step 4.1

Factor $5 x + 7$ out of $(x + 1) (5 x + 7)$ .

$\frac{(x - 3) (2 x + 7)}{(5 x + 7) (x + 1)} \cdot \frac{5 x + 7}{3 x - 9}$

Step 4.2

Cancel the common factor.

$\frac{(x - 3) (2 x + 7)}{(5 x + 7) (x + 1)} \cdot \frac{5 x + 7}{3 x - 9}$

Step 4.3

Rewrite the expression.

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

Step 5

Multiply $\frac{(x - 3) (2 x + 7)}{x + 1}$ by $\frac{1}{3 x - 9}$ .

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

Step 6

Factor $3$ out of $3 x - 9$ .

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Step 6.1

Factor $3$ out of $3 x$ .

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

Step 6.2

Factor $3$ out of $- 9$ .

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

Step 6.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

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

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

Step 7

Cancel the common factor of $x - 3$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Move $3$ to the left of $x + 1$ .

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