Algebra Examples

$\frac{9 x^{2} - 9 x}{3 x - 3} \cdot \frac{6 x^{2} + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $9 x$ out of $9 x^{2} - 9 x$ .

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Step 1.1.1

Factor $9 x$ out of $9 x^{2}$ .

$\frac{9 x (x) - 9 x}{3 x - 3} \cdot \frac{6 x^{2} + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 1.1.2

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

$\frac{9 x (x) + 9 x (- 1)}{3 x - 3} \cdot \frac{6 x^{2} + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 1.1.3

Factor $9 x$ out of $9 x (x) + 9 x (- 1)$ .

$\frac{9 x (x - 1)}{3 x - 3} \cdot \frac{6 x^{2} + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 1.2

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

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Step 1.2.1

Factor $3$ out of $3 x$ .

$\frac{9 x (x - 1)}{3 (x) - 3} \cdot \frac{6 x^{2} + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 1.2.2

Factor $3$ out of $- 3$ .

$\frac{9 x (x - 1)}{3 (x) + 3 (- 1)} \cdot \frac{6 x^{2} + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 1.2.3

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

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{6 x^{2} + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $6 x^{2} + 21 x - 27$ .

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Step 2.1.1

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

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2}) + 21 x - 27}{2 x^{2} + 13 x + 18}$

Step 2.1.2

Factor $3$ out of $21 x$ .

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2}) + 3 (7 x) - 27}{2 x^{2} + 13 x + 18}$

Step 2.1.3

Factor $3$ out of $- 27$ .

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2}) + 3 (7 x) + 3 (- 9)}{2 x^{2} + 13 x + 18}$

Step 2.1.4

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

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2} + 7 x) + 3 (- 9)}{2 x^{2} + 13 x + 18}$

Step 2.1.5

Factor $3$ out of $3 (2 x^{2} + 7 x) + 3 (- 9)$ .

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2} + 7 x - 9)}{2 x^{2} + 13 x + 18}$

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 $a \cdot c = 2 \cdot - 9 = - 18$ and whose sum is $b = 7$ .

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Step 2.2.1.1

Factor $7$ out of $7 x$ .

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2} + 7 (x) - 9)}{2 x^{2} + 13 x + 18}$

Step 2.2.1.2

Rewrite $7$ as $- 2$ plus $9$

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2} + (- 2 + 9) x - 9)}{2 x^{2} + 13 x + 18}$

Step 2.2.1.3

Apply the distributive property.

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x^{2} - 2 x + 9 x - 9)}{2 x^{2} + 13 x + 18}$

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.

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 ((2 x^{2} - 2 x) + 9 x - 9)}{2 x^{2} + 13 x + 18}$

Step 2.2.2.2

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

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (2 x (x - 1) + 9 (x - 1))}{2 x^{2} + 13 x + 18}$

Step 2.2.3

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

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 ((x - 1) (2 x + 9))}{2 x^{2} + 13 x + 18}$

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (x - 1) (2 x + 9)}{2 x^{2} + 13 x + 18}$

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 $a \cdot c = 2 \cdot 18 = 36$ and whose sum is $b = 13$ .

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Step 3.1.1

Factor $13$ out of $13 x$ .

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (x - 1) (2 x + 9)}{2 x^{2} + 13 (x) + 18}$

Step 3.1.2

Rewrite $13$ as $4$ plus $9$

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (x - 1) (2 x + 9)}{2 x^{2} + (4 + 9) x + 18}$

Step 3.1.3

Apply the distributive property.

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (x - 1) (2 x + 9)}{2 x^{2} + 4 x + 9 x + 18}$

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.

$\frac{9 x (x - 1)}{3 (x - 1)} \cdot \frac{3 (x - 1) (2 x + 9)}{(2 x^{2} + 4 x) + 9 x + 18}$

Step 3.2.2

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

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

Step 3.3

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

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

Step 4

Cancel the common factor of $3 (x - 1)$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Combine $9$ and $\frac{2 x + 9}{(x + 2) (2 x + 9)}$ .

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

Step 6

Combine $\frac{9 (2 x + 9)}{(x + 2) (2 x + 9)}$ and $x$ .

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

Step 7

Cancel the common factor of $2 x + 9$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Multiply $\frac{9 x}{x + 2}$ by $x - 1$ .

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