Problem 19 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x^{2}-5 x+6}{x^{2}-2 x-3} \cdot \frac{x^{2}-1}{x^{2}-4}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \[((x-3)(x+1))/{((x+3)(x+2))}\].

Step 1: Factorize each Polynomial

In order to factorize the polynomial \(x^{2}-5 x+6\), this can be broken down into two binomial expressions \((x-2)(x-3)\). Similarly, the polynomial \(x^{2}-2 x-3\) can be factored into \((x-1)(x+3)\), \(x^{2}-1\) can be factored as \((x+1)(x-1)\), and finally, \(x^{2}-4\) can be factored as \((x+2)(x-2)\).

Step 2: Substitute factored expressions into original expression

Replace each polynomial in the given expression with their respective factored form. Gives us the following equation, \[\frac{(x-2)(x-3)}{(x-1)(x+3)} \cdot \frac{(x+1)(x-1)}{(x+2)(x-2)} \]

Step 3: Cancel out common expressions

Simplify the expression by cancelling out common binomial expressions from the numerator and the denominator. \(x-2\) and \(x-1\) are present in both the numerator and the denominator. Therefore, the simplified expression becomes \[\frac{(x-3)}{(x+3)} \cdot \frac{(x+1)}{(x+2)}.\]

Step 4: Multiply factored expressions

Multiply the resulting factors to get the final simplified form of the given expression, \[((x-3)(x+1))/{((x+3)(x+2))}.\]