Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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Step 6.2.1

Move $x$ .

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

Step 6.2.2

Multiply $x$ by $x$ .

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

Step 6.3

Multiply $- 1$ by $1$ .

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

Step 6.4

Subtract $x$ from $x$ .

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

Step 6.5

Add $- 3 - x^{2}$ and $0$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 7.2

Factor $- 1$ out of $- x^{2}$ .

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

Step 7.3

Factor $- 1$ out of $- 1 (3) - (x^{2})$ .

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

Step 7.4

Move the negative in front of the fraction.

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