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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 expression after multiplication and division is \(\frac{x-1}{x+2}\).

Step by step solution

01

Factorize the Quadratic Expressions

Start by factorizing all the quadratic expressions in the expression. \(x^{2}-5x+6\) can be written as \((x-2)(x-3)\), \(x^{2}-2x-3\) as \((x-3)(x+1)\), \(x^{2}-1\) as \((x-1)(x+1)\), and \(x^{2}-4\) as \((x-2)(x+2)\).
02

Rewrite the Expression with Factors

Rewrite the given expression with these factors: \[\frac{(x-2)(x-3)}{(x-3)(x+1)} \cdot \frac{(x-1)(x+1)}{(x-2)(x+2)}\]
03

Cancel out Common Factors

From the expression, we can see that \(x-2\), \(x-3\), \(x+1\) are common in both the numerator and denominator of the fractions. Therefore, cancel out these common factors, and we are left with \(x-1\) in the numerator and \(x+2\) in the denominator.
04

Write the final simplified Expression

After cancellation, the final simplified form of the expression is \[\frac{x-1}{x+2}\]

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x,\) is modeled by \(|x-20| \leq 5\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(a x^{2}+c=0, a \neq 0,\) cannot be solved by the quadratic formula.

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6} $$

The rational expression $$\frac{130 x}{100-x}$$ describes the cost, in millions of dollars, to inoculate \(x\) percent of the population against a particular strain of flu. a. Evaluate the expression for \(x=40, x=80,\) and \(x=90\) Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

Perform the indicated operations. $$\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right)$$
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