Simplify (5x-1)/(6x^2+13x-5)-(5x-1)/(12x^2-x-1)

$\frac{5 x - 1}{6 x^{2} + 13 x - 5} - \frac{5 x - 1}{12 x^{2} - x - 1}$

Step 1

Simplify each term.

Factor by grouping.

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 $a \cdot c = 6 \cdot - 5 = - 30$ and whose sum is $b = 13$ .

Factor $13$ out of $13 x$ .

$\frac{5 x - 1}{6 x^{2} + 13 (x) - 5} - \frac{5 x - 1}{12 x^{2} - x - 1}$

Rewrite $13$ as $- 2$ plus $15$

$\frac{5 x - 1}{6 x^{2} + (- 2 + 15) x - 5} - \frac{5 x - 1}{12 x^{2} - x - 1}$

Apply the distributive property.

$\frac{5 x - 1}{6 x^{2} - 2 x + 15 x - 5} - \frac{5 x - 1}{12 x^{2} - x - 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{5 x - 1}{(6 x^{2} - 2 x) + 15 x - 5} - \frac{5 x - 1}{12 x^{2} - x - 1}$

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

$\frac{5 x - 1}{2 x (3 x - 1) + 5 (3 x - 1)} - \frac{5 x - 1}{12 x^{2} - x - 1}$

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

$\frac{5 x - 1}{(3 x - 1) (2 x + 5)} - \frac{5 x - 1}{12 x^{2} - x - 1}$

Factor by grouping.

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

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

$\frac{5 x - 1}{(3 x - 1) (2 x + 5)} - \frac{5 x - 1}{12 x^{2} - (x) - 1}$

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

$\frac{5 x - 1}{(3 x - 1) (2 x + 5)} - \frac{5 x - 1}{12 x^{2} + (3 - 4) x - 1}$

Apply the distributive property.

$\frac{5 x - 1}{(3 x - 1) (2 x + 5)} - \frac{5 x - 1}{12 x^{2} + 3 x - 4 x - 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{5 x - 1}{(3 x - 1) (2 x + 5)} - \frac{5 x - 1}{(12 x^{2} + 3 x) - 4 x - 1}$

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

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

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

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

Step 2

To write $\frac{5 x - 1}{(3 x - 1) (2 x + 5)}$ as a fraction with a common denominator, multiply by $\frac{4 x + 1}{4 x + 1}$ .

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

Step 3

To write $- \frac{5 x - 1}{(4 x + 1) (3 x - 1)}$ as a fraction with a common denominator, multiply by $\frac{2 x + 5}{2 x + 5}$ .

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

Step 4

Write each expression with a common denominator of $(3 x - 1) (2 x + 5) (4 x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(3 x - 1) (2 x + 5) (4 x + 1)$ .

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

Reorder the factors of $(4 x + 1) (3 x - 1) (2 x + 5)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Factor $5 x - 1$ out of $(5 x - 1) (4 x + 1) - (5 x - 1) (2 x + 5)$ .

Factor $5 x - 1$ out of $- (5 x - 1) (2 x + 5)$ .

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

Factor $5 x - 1$ out of $(5 x - 1) (4 x + 1) + (5 x - 1) (- 1 (2 x + 5))$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $5$ .

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

Subtract $2 x$ from $4 x$ .

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

Subtract $5$ from $1$ .

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

Factor $2$ out of $2 x - 4$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 4$ .

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

Factor $2$ out of $2 x + 2 \cdot - 2$ .

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

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

Step 7

Move $2$ to the left of $5 x - 1$ .

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

Simplify (5x-1)/(6x^2+13x-5)-(5x-1)/(12x^2-x-1)

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