Precalculus Examples

$\frac{5}{2 x - 8} + \frac{3 x^{2} + 15}{x^{3} - 4 x^{2} + 5 x - 20}$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $2 x - 8$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{5}{2 (x) - 8} + \frac{3 x^{2} + 15}{x^{3} - 4 x^{2} + 5 x - 20}$

Step 1.1.2

Factor $2$ out of $- 8$ .

$\frac{5}{2 x + 2 \cdot - 4} + \frac{3 x^{2} + 15}{x^{3} - 4 x^{2} + 5 x - 20}$

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot - 4$ .

$\frac{5}{2 (x - 4)} + \frac{3 x^{2} + 15}{x^{3} - 4 x^{2} + 5 x - 20}$

Step 1.2

Factor $3$ out of $3 x^{2} + 15$ .

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Step 1.2.1

Factor $3$ out of $3 x^{2}$ .

$\frac{5}{2 (x - 4)} + \frac{3 (x^{2}) + 15}{x^{3} - 4 x^{2} + 5 x - 20}$

Step 1.2.2

Factor $3$ out of $15$ .

$\frac{5}{2 (x - 4)} + \frac{3 x^{2} + 3 \cdot 5}{x^{3} - 4 x^{2} + 5 x - 20}$

Step 1.2.3

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

$\frac{5}{2 (x - 4)} + \frac{3 (x^{2} + 5)}{x^{3} - 4 x^{2} + 5 x - 20}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Factor out the greatest common factor from each group.

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Step 1.3.1.1

Group the first two terms and the last two terms.

$\frac{5}{2 (x - 4)} + \frac{3 (x^{2} + 5)}{(x^{3} - 4 x^{2}) + 5 x - 20}$

Step 1.3.1.2

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

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

Step 1.3.2

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

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

Step 1.4

Cancel the common factor of $x^{2} + 5$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Reorder the factors of $(x - 4) \cdot 2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $3$ by $2$ .

$\frac{5 + 6}{2 (x - 4)}$

Step 5.2

Add $5$ and $6$ .

$\frac{11}{2 (x - 4)}$