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Chapter 0: Problem 19

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

Short Answer

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

Step by step solution

01

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 \((x+2)(x-2)\).
02

Substitute factored expressions into original expression

Replace each polynomial in the given expression with their respective factored form. Gives us the following equation, \[\frac{(x-2)(x-3)}{(x-1)(x+3)} \cdot \frac{(x+1)(x-1)}{(x+2)(x-2)} \]
03

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 \[\frac{(x-3)}{(x+3)} \cdot \frac{(x+1)}{(x+2)}.\]
04

Multiply factored expressions

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

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Most popular questions from this chapter

$$\frac{5}{4} \cdot \frac{8}{15}$$

Determine whether each statement in Exercises 43鈥�50 is true or false. $$-3>-13$$

Explain how to factor the difference of two squares. Provide an example with your explanation.

Evaluate each algebraic expression for the given value or values of the variable(s). $$\frac{2 x+y}{x y-2 x}, \text { for } x=-2 \text { and } y=4$$

Fill in each box to make the statement true. Find the exact value of \(\sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}}\) without the use of a calculator.
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