Problem 17 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \((x-3)*(x+3) / x^{2}\) * \(x*(x-3) / ((x-4)*(x+3))\).

Step 2: 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 \(x / (x*(x-4))\).

Step 3: 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.