Pre-Algebra Examples

$\frac{2 x - 1}{6 x^{3} + 5 x^{2} + 6 x - 5 x}$

Step 1

Simplify the denominator.

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Step 1.1

Subtract $5 x$ from $6 x$ .

$\frac{2 x - 1}{6 x^{3} + 5 x^{2} + x}$

Step 1.2

Factor $x$ out of $6 x^{3} + 5 x^{2} + x$ .

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Step 1.2.1

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

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

Step 1.2.2

Factor $x$ out of $5 x^{2}$ .

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

Step 1.2.3

Raise $x$ to the power of $1$ .

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

Step 1.2.4

Factor $x$ out of $x^{1}$ .

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

Step 1.2.5

Factor $x$ out of $x (6 x^{2}) + x (5 x)$ .

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

Step 1.2.6

Factor $x$ out of $x (6 x^{2} + 5 x) + x \cdot 1$ .

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

Step 1.3

Factor by grouping.

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Step 1.3.1

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 $a \cdot c = 6 \cdot 1 = 6$ and whose sum is $b = 5$ .

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Step 1.3.1.1

Factor $5$ out of $5 x$ .

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

Step 1.3.1.2

Rewrite $5$ as $2$ plus $3$

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

Step 1.3.1.3

Apply the distributive property.

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

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

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

Step 1.3.2.2

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

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

Step 1.3.3

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

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

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

Step 2

Split the fraction $\frac{2 x - 1}{x (3 x + 1) (2 x + 1)}$ into two fractions.

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

Step 3

Cancel the common factor of $x$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Move the negative in front of the fraction.

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