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Chapter 0: Problem 17

Multiply or divide as indicated. $$\frac{x^{2}-9}{x^{2}} \cdot \frac{x^{2}-3 x}{x^{2}+x-12}$$

Short Answer

Expert verified
The simplified form of the given expression is \(1 / (x-4)\).

Step by step solution

01

Factorize both parts of the fraction

We start by factoring the given expressions. The first fraction \((x^{2} - 9) / x^{2}\) can be factored as \((x-3)*(x+3) / x^{2}\) because \(x^{2} - 9\) is the difference of squares. For the second fraction \((x^{2} - 3x) / (x^{2} + x - 12)\), we can factorize it as \(x*(x-3) / ((x-4)*(x+3))\). The resulting expression is then \((x-3)*(x+3) / x^{2}\) * \(x*(x-3) / ((x-4)*(x+3))\).
02

Simplify the result

Now we can simplify the expression by cancelling out the common factors from the numerator and the denominator. Here, the common factors are \((x+3)\) and \((x-3)\). After cancelling out these factors, we get \(x / (x*(x-4))\).
03

Further simplify the expression

In the expression \(x / (x*(x-4))\), we can cancel out \(x\), resulting in \(1 / (x-4)\). Whether \(x\) can be cancelled depends on if \(x\) can be 0. Normally, we need to remember not to cancel factors if their removal would allow for a division by zero. Here, it is implied from the initial problem that \(x\) cannot be zero.

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Most popular questions from this chapter

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

$$\text { Factor completely.}$$ $$12 x^{2}(x-1)-4 x(x-1)-5(x-1)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. . You grouped the polynomial's terms using different groupings than I did, yet we both obtained the same factorization.

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$

A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula $$h=4+60 t-16 t^{2}$$ describes the ball's height above the ground, \(h,\) in feet, \(t\) seconds after it was kicked. Use this formula to solve Exercises \(19-20 .\) What was the ball鈥檚 height 3 seconds after it was kicked?
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