Precalculus Examples

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

Step 1

Factor $x^{2} + 2 x - 3$ using the AC method.

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Step 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Reorder terms.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 3.3.1

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

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

Step 3.3.2

Factor $- 1$ out of $- x$ .

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

Step 3.3.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

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

Step 3.3.4

Reorder terms.

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

Step 3.3.5

Cancel the common factor.

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

Step 3.3.6

Rewrite the expression.

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Move $- 1$ to the left of $x - 1$ .

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

Step 3.4.2

Move the negative in front of the fraction.

$- \frac{x - 1}{x + 1}$