Algebra Examples

$\frac{2 x^{2} + 7 x - 15}{3 x^{2} + 16 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = 7$ .

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Step 1.1.1

Factor $7$ out of $7 x$ .

$\frac{2 x^{2} + 7 (x) - 15}{3 x^{2} + 16 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 1.1.2

Rewrite $7$ as $- 3$ plus $10$

$\frac{2 x^{2} + (- 3 + 10) x - 15}{3 x^{2} + 16 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 1.1.3

Apply the distributive property.

$\frac{2 x^{2} - 3 x + 10 x - 15}{3 x^{2} + 16 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 3 x) + 10 x - 15}{3 x^{2} + 16 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 1.2.2

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

$\frac{x (2 x - 3) + 5 (2 x - 3)}{3 x^{2} + 16 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 1.3

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

$\frac{(2 x - 3) (x + 5)}{3 x^{2} + 16 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

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 $a \cdot c = 3 \cdot 5 = 15$ and whose sum is $b = 16$ .

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Step 2.1.1

Factor $16$ out of $16 x$ .

$\frac{(2 x - 3) (x + 5)}{3 x^{2} + 16 (x) + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 2.1.2

Rewrite $16$ as $1$ plus $15$

$\frac{(2 x - 3) (x + 5)}{3 x^{2} + (1 + 15) x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 2.1.3

Apply the distributive property.

$\frac{(2 x - 3) (x + 5)}{3 x^{2} + 1 x + 15 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 2.1.4

Multiply $x$ by $1$ .

$\frac{(2 x - 3) (x + 5)}{3 x^{2} + x + 15 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

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.

$\frac{(2 x - 3) (x + 5)}{(3 x^{2} + x) + 15 x + 5} \cdot \frac{3 x^{2} - 2 x - 1}{2 x^{2} - x - 3}$

Step 2.2.2

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

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

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

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

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

Multiply $x$ by $1$ .

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

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.

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

Step 3.2.2

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

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

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

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Step 4.1.1

Factor $- 1$ out of $- x$ .

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

Step 4.1.2

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

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

Step 4.1.3

Apply the distributive property.

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

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.

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $3 x + 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{x + 5}{x + 5} \cdot \frac{x - 1}{x + 1}$

Step 7

Multiply $\frac{x + 5}{x + 5}$ by $\frac{x - 1}{x + 1}$ .

$\frac{(x + 5) (x - 1)}{(x + 5) (x + 1)}$

Step 8

Cancel the common factor of $x + 5$ .

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Step 8.1

Cancel the common factor.

$\frac{(x + 5) (x - 1)}{(x + 5) (x + 1)}$

Step 8.2

Rewrite the expression.

$\frac{x - 1}{x + 1}$