Calculus Examples

$\frac{2 x^{3} - x^{2} - 6 x}{2 x^{2} - 7 x + 6 x}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{x (2 x^{2}) - x^{2} - 6 x}{2 x^{2} - 7 x + 6 x}$

Step 1.1.2

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

$\frac{x (2 x^{2}) + x (- x) - 6 x}{2 x^{2} - 7 x + 6 x}$

Step 1.1.3

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

$\frac{x (2 x^{2}) + x (- x) + x \cdot - 6}{2 x^{2} - 7 x + 6 x}$

Step 1.1.4

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

$\frac{x (2 x^{2} - x) + x \cdot - 6}{2 x^{2} - 7 x + 6 x}$

Step 1.1.5

Factor $x$ out of $x (2 x^{2} - x) + x \cdot - 6$ .

$\frac{x (2 x^{2} - x - 6)}{2 x^{2} - 7 x + 6 x}$

Step 1.2

Factor by grouping.

Tap for more steps...

Step 1.2.1

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 1$ .

Tap for more steps...

Step 1.2.1.1

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

$\frac{x (2 x^{2} - (x) - 6)}{2 x^{2} - 7 x + 6 x}$

Step 1.2.1.2

Rewrite $- 1$ as $3$ plus $- 4$

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

Step 1.2.1.3

Apply the distributive property.

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

Step 1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2.2

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

$\frac{x (x (2 x + 3) - 2 (2 x + 3))}{2 x^{2} - 7 x + 6 x}$

Step 1.2.3

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

$\frac{x ((2 x + 3) (x - 2))}{2 x^{2} - 7 x + 6 x}$

$\frac{x (2 x + 3) (x - 2)}{2 x^{2} - 7 x + 6 x}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Add $- 7 x$ and $6 x$ .

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Factor $x$ out of $- x$ .

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

Step 2.2.3

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

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

Step 3

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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