Precalculus Examples

$\frac{6}{5 x^{2} + 26 x + 5} - \frac{1}{x^{2} + x - 20}$

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 $a \cdot c = 5 \cdot 5 = 25$ and whose sum is $b = 26$ .

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Step 1.1.1.1

Factor $26$ out of $26 x$ .

$\frac{6}{5 x^{2} + 26 (x) + 5} - \frac{1}{x^{2} + x - 20}$

Step 1.1.1.2

Rewrite $26$ as $1$ plus $25$

$\frac{6}{5 x^{2} + (1 + 25) x + 5} - \frac{1}{x^{2} + x - 20}$

Step 1.1.1.3

Apply the distributive property.

$\frac{6}{5 x^{2} + 1 x + 25 x + 5} - \frac{1}{x^{2} + x - 20}$

Step 1.1.1.4

Multiply $x$ by $1$ .

$\frac{6}{5 x^{2} + x + 25 x + 5} - \frac{1}{x^{2} + x - 20}$

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.

$\frac{6}{(5 x^{2} + x) + 25 x + 5} - \frac{1}{x^{2} + x - 20}$

Step 1.1.2.2

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

$\frac{6}{x (5 x + 1) + 5 (5 x + 1)} - \frac{1}{x^{2} + x - 20}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 1$ .

$\frac{6}{(5 x + 1) (x + 5)} - \frac{1}{x^{2} + x - 20}$

Step 1.2

Factor $x^{2} + x - 20$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{6}{(5 x + 1) (x + 5)} - \frac{1}{(x - 4) (x + 5)}$

Step 2

To write $\frac{6}{(5 x + 1) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{6}{(5 x + 1) (x + 5)} \cdot \frac{x - 4}{x - 4} - \frac{1}{(x - 4) (x + 5)}$

Step 3

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

$\frac{6}{(5 x + 1) (x + 5)} \cdot \frac{x - 4}{x - 4} - \frac{1}{(x - 4) (x + 5)} \cdot \frac{5 x + 1}{5 x + 1}$

Step 4

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

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Step 4.1

Multiply $\frac{6}{(5 x + 1) (x + 5)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{6 (x - 4)}{(5 x + 1) (x + 5) (x - 4)} - \frac{1}{(x - 4) (x + 5)} \cdot \frac{5 x + 1}{5 x + 1}$

Step 4.2

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

$\frac{6 (x - 4)}{(5 x + 1) (x + 5) (x - 4)} - \frac{5 x + 1}{(x - 4) (x + 5) (5 x + 1)}$

Step 4.3

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

$\frac{6 (x - 4)}{(5 x + 1) (x + 5) (x - 4)} - \frac{5 x + 1}{(5 x + 1) (x + 5) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 (x - 4) - (5 x + 1)}{(5 x + 1) (x + 5) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{6 x + 6 \cdot - 4 - (5 x + 1)}{(5 x + 1) (x + 5) (x - 4)}$

Step 6.2

Multiply $6$ by $- 4$ .

$\frac{6 x - 24 - (5 x + 1)}{(5 x + 1) (x + 5) (x - 4)}$

Step 6.3

Apply the distributive property.

$\frac{6 x - 24 - (5 x) - 1 \cdot 1}{(5 x + 1) (x + 5) (x - 4)}$

Step 6.4

Multiply $5$ by $- 1$ .

$\frac{6 x - 24 - 5 x - 1 \cdot 1}{(5 x + 1) (x + 5) (x - 4)}$

Step 6.5

Multiply $- 1$ by $1$ .

$\frac{6 x - 24 - 5 x - 1}{(5 x + 1) (x + 5) (x - 4)}$

Step 6.6

Subtract $5 x$ from $6 x$ .

$\frac{x - 24 - 1}{(5 x + 1) (x + 5) (x - 4)}$

Step 6.7

Subtract $1$ from $- 24$ .

$\frac{x - 25}{(5 x + 1) (x + 5) (x - 4)}$