Algebra Examples

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

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 $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = 2$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

Apply the distributive property.

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

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.

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Multiply $\frac{1}{x - 1}$ by $\frac{3 x + 5}{3 x + 5}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Add $2$ and $5$ .

$\frac{3 x + 7}{(x - 1) (3 x + 5)}$