Precalculus Examples

$\frac{9 x + 2}{3 x^{2} - 2 x - 8} + \frac{7}{3 x^{2} + x - 4}$

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 1.1.1.1

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

$\frac{9 x + 2}{3 x^{2} - 2 (x) - 8} + \frac{7}{3 x^{2} + x - 4}$

Step 1.1.1.2

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

$\frac{9 x + 2}{3 x^{2} + (4 - 6) x - 8} + \frac{7}{3 x^{2} + x - 4}$

Step 1.1.1.3

Apply the distributive property.

$\frac{9 x + 2}{3 x^{2} + 4 x - 6 x - 8} + \frac{7}{3 x^{2} + x - 4}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{9 x + 2}{(3 x^{2} + 4 x) - 6 x - 8} + \frac{7}{3 x^{2} + x - 4}$

Step 1.1.2.2

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

$\frac{9 x + 2}{x (3 x + 4) - 2 (3 x + 4)} + \frac{7}{3 x^{2} + x - 4}$

Step 1.1.3

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

$\frac{9 x + 2}{(3 x + 4) (x - 2)} + \frac{7}{3 x^{2} + x - 4}$

Step 1.2

Factor by grouping.

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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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

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Step 1.2.1.1

Multiply by $1$ .

$\frac{9 x + 2}{(3 x + 4) (x - 2)} + \frac{7}{3 x^{2} + 1 x - 4}$

Step 1.2.1.2

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

$\frac{9 x + 2}{(3 x + 4) (x - 2)} + \frac{7}{3 x^{2} + (- 3 + 4) x - 4}$

Step 1.2.1.3

Apply the distributive property.

$\frac{9 x + 2}{(3 x + 4) (x - 2)} + \frac{7}{3 x^{2} - 3 x + 4 x - 4}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{9 x + 2}{(3 x + 4) (x - 2)} + \frac{7}{(3 x^{2} - 3 x) + 4 x - 4}$

Step 1.2.2.2

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

$\frac{9 x + 2}{(3 x + 4) (x - 2)} + \frac{7}{3 x (x - 1) + 4 (x - 1)}$

Step 1.2.3

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

$\frac{9 x + 2}{(3 x + 4) (x - 2)} + \frac{7}{(x - 1) (3 x + 4)}$

Step 2

To write $\frac{9 x + 2}{(3 x + 4) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{9 x + 2}{(3 x + 4) (x - 2)} \cdot \frac{x - 1}{x - 1} + \frac{7}{(x - 1) (3 x + 4)}$

Step 3

To write $\frac{7}{(x - 1) (3 x + 4)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{9 x + 2}{(3 x + 4) (x - 2)} \cdot \frac{x - 1}{x - 1} + \frac{7}{(x - 1) (3 x + 4)} \cdot \frac{x - 2}{x - 2}$

Step 4

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

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Step 4.1

Multiply $\frac{9 x + 2}{(3 x + 4) (x - 2)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{(9 x + 2) (x - 1)}{(3 x + 4) (x - 2) (x - 1)} + \frac{7}{(x - 1) (3 x + 4)} \cdot \frac{x - 2}{x - 2}$

Step 4.2

Multiply $\frac{7}{(x - 1) (3 x + 4)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{(9 x + 2) (x - 1)}{(3 x + 4) (x - 2) (x - 1)} + \frac{7 (x - 2)}{(x - 1) (3 x + 4) (x - 2)}$

Step 4.3

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

$\frac{(9 x + 2) (x - 1)}{(3 x + 4) (x - 1) (x - 2)} + \frac{7 (x - 2)}{(x - 1) (3 x + 4) (x - 2)}$

Step 4.4

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

$\frac{(9 x + 2) (x - 1)}{(3 x + 4) (x - 1) (x - 2)} + \frac{7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(9 x + 2) (x - 1) + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(9 x + 2) (x - 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{9 x (x - 1) + 2 (x - 1) + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.1.2

Apply the distributive property.

$\frac{9 x \cdot x + 9 x \cdot - 1 + 2 (x - 1) + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.1.3

Apply the distributive property.

$\frac{9 x \cdot x + 9 x \cdot - 1 + 2 x + 2 \cdot - 1 + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1.1.1

Move $x$ .

$\frac{9 (x \cdot x) + 9 x \cdot - 1 + 2 x + 2 \cdot - 1 + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.2.1.1.2

Multiply $x$ by $x$ .

$\frac{9 x^{2} + 9 x \cdot - 1 + 2 x + 2 \cdot - 1 + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.2.1.2

Multiply $- 1$ by $9$ .

$\frac{9 x^{2} - 9 x + 2 x + 2 \cdot - 1 + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.2.1.3

Multiply $2$ by $- 1$ .

$\frac{9 x^{2} - 9 x + 2 x - 2 + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.2.2

Add $- 9 x$ and $2 x$ .

$\frac{9 x^{2} - 7 x - 2 + 7 (x - 2)}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.3

Apply the distributive property.

$\frac{9 x^{2} - 7 x - 2 + 7 x + 7 \cdot - 2}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.4

Multiply $7$ by $- 2$ .

$\frac{9 x^{2} - 7 x - 2 + 7 x - 14}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.5

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

$\frac{9 x^{2} + 0 - 2 - 14}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.6

Add $9 x^{2}$ and $0$ .

$\frac{9 x^{2} - 2 - 14}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.7

Subtract $14$ from $- 2$ .

$\frac{9 x^{2} - 16}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.8

Rewrite $9 x^{2} - 16$ in a factored form.

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Step 6.8.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{{(3 x)}^{2} - 16}{(3 x + 4) (x - 1) (x - 2)}$

Step 6.8.2

Rewrite $16$ as $4^{2}$ .

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

Step 6.8.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 4$ .

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

Step 7

Cancel the common factor of $3 x + 4$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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