Precalculus Examples

$\frac{2}{3 x^{2} + 2 x - 8} + \frac{1}{x + 2}$

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = 2$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{2}{3 x^{2} + 2 (x) - 8} + \frac{1}{x + 2}$

Step 1.1.2

Rewrite $2$ as $- 4$ plus $6$

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

Step 1.1.3

Apply the distributive property.

$\frac{2}{3 x^{2} - 4 x + 6 x - 8} + \frac{1}{x + 2}$

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.

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

Step 1.2.2

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

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

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 4$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Subtract $4$ from $2$ .

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