Algebra Examples

$\frac{\frac{4 x^{3} - 12 x^{2}}{4 x^{2} + 7 x - 2}}{\frac{2 x^{2} - 6 x}{5 x^{2} + 11 x + 2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 x^{3} - 12 x^{2}}{4 x^{2} + 7 x - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 2

Factor $4 x^{2}$ out of $4 x^{3} - 12 x^{2}$ .

Tap for more steps...

Step 2.1

Factor $4 x^{2}$ out of $4 x^{3}$ .

$\frac{4 x^{2} (x) - 12 x^{2}}{4 x^{2} + 7 x - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 2.2

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

$\frac{4 x^{2} (x) + 4 x^{2} (- 3)}{4 x^{2} + 7 x - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 2.3

Factor $4 x^{2}$ out of $4 x^{2} (x) + 4 x^{2} (- 3)$ .

$\frac{4 x^{2} (x - 3)}{4 x^{2} + 7 x - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 3

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is $b = 7$ .

Tap for more steps...

Step 3.1.1

Factor $7$ out of $7 x$ .

$\frac{4 x^{2} (x - 3)}{4 x^{2} + 7 (x) - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 3.1.2

Rewrite $7$ as $- 1$ plus $8$

$\frac{4 x^{2} (x - 3)}{4 x^{2} + (- 1 + 8) x - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 3.1.3

Apply the distributive property.

$\frac{4 x^{2} (x - 3)}{4 x^{2} - 1 x + 8 x - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$\frac{4 x^{2} (x - 3)}{(4 x^{2} - 1 x) + 8 x - 2} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 3.2.2

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

$\frac{4 x^{2} (x - 3)}{x (4 x - 1) + 2 (4 x - 1)} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{5 x^{2} + 11 x + 2}{2 x^{2} - 6 x}$

Step 4

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 5 \cdot 2 = 10$ and whose sum is $b = 11$ .

Tap for more steps...

Step 4.1.1

Factor $11$ out of $11 x$ .

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{5 x^{2} + 11 (x) + 2}{2 x^{2} - 6 x}$

Step 4.1.2

Rewrite $11$ as $1$ plus $10$

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{5 x^{2} + (1 + 10) x + 2}{2 x^{2} - 6 x}$

Step 4.1.3

Apply the distributive property.

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{5 x^{2} + 1 x + 10 x + 2}{2 x^{2} - 6 x}$

Step 4.1.4

Multiply $x$ by $1$ .

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{5 x^{2} + x + 10 x + 2}{2 x^{2} - 6 x}$

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{(5 x^{2} + x) + 10 x + 2}{2 x^{2} - 6 x}$

Step 4.2.2

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

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{x (5 x + 1) + 2 (5 x + 1)}{2 x^{2} - 6 x}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 1$ .

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{(5 x + 1) (x + 2)}{2 x^{2} - 6 x}$

Step 5

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

Tap for more steps...

Step 5.1

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

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{(5 x + 1) (x + 2)}{2 x (x) - 6 x}$

Step 5.2

Factor $2 x$ out of $- 6 x$ .

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{(5 x + 1) (x + 2)}{2 x (x) + 2 x (- 3)}$

Step 5.3

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

$\frac{4 x^{2} (x - 3)}{(4 x - 1) (x + 2)} \cdot \frac{(5 x + 1) (x + 2)}{2 x (x - 3)}$

Step 6

Combine.

$\frac{4 x^{2} (x - 3) ((5 x + 1) (x + 2))}{(4 x - 1) (x + 2) (2 x (x - 3))}$

Step 7

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 7.1

Factor $2$ out of $4 x^{2} (x - 3) ((5 x + 1) (x + 2))$ .

$\frac{2 (2 x^{2} (x - 3) ((5 x + 1) (x + 2)))}{(4 x - 1) (x + 2) (2 x (x - 3))}$

Step 7.2

Cancel the common factors.

Tap for more steps...

Step 7.2.1

Factor $2$ out of $(4 x - 1) (x + 2) (2 x (x - 3))$ .

$\frac{2 (2 x^{2} (x - 3) ((5 x + 1) (x + 2)))}{2 ((4 x - 1) (x + 2) (x (x - 3)))}$

Step 7.2.2

Cancel the common factor.

$\frac{2 (2 x^{2} (x - 3) ((5 x + 1) (x + 2)))}{2 ((4 x - 1) (x + 2) (x (x - 3)))}$

Step 7.2.3

Rewrite the expression.

$\frac{2 x^{2} (x - 3) ((5 x + 1) (x + 2))}{(4 x - 1) (x + 2) (x (x - 3))}$

Step 8

Cancel the common factor of $x^{2}$ and $x$ .

Tap for more steps...

Step 8.1

Factor $x$ out of $2 x^{2} (x - 3) ((5 x + 1) (x + 2))$ .

$\frac{x (2 x (x - 3) ((5 x + 1) (x + 2)))}{(4 x - 1) (x + 2) (x (x - 3))}$

Step 8.2

Cancel the common factors.

Tap for more steps...

Step 8.2.1

Factor $x$ out of $(4 x - 1) (x + 2) (x (x - 3))$ .

$\frac{x (2 x (x - 3) ((5 x + 1) (x + 2)))}{x ((4 x - 1) (x + 2) (x - 3))}$

Step 8.2.2

Cancel the common factor.

$\frac{x (2 x (x - 3) ((5 x + 1) (x + 2)))}{x ((4 x - 1) (x + 2) (x - 3))}$

Step 8.2.3

Rewrite the expression.

$\frac{2 x (x - 3) ((5 x + 1) (x + 2))}{(4 x - 1) (x + 2) (x - 3)}$

Step 9

Cancel the common factor of $x - 3$ .

Tap for more steps...

Step 9.1

Cancel the common factor.

$\frac{2 x (x - 3) ((5 x + 1) (x + 2))}{(4 x - 1) (x + 2) (x - 3)}$

Step 9.2

Rewrite the expression.

$\frac{2 x ((5 x + 1) (x + 2))}{(4 x - 1) (x + 2)}$

Step 10

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 10.1

Cancel the common factor.

$\frac{2 x ((5 x + 1) (x + 2))}{(4 x - 1) (x + 2)}$

Step 10.2

Rewrite the expression.

$\frac{2 x (5 x + 1)}{4 x - 1}$