Problem 11 - Introductory Algebra for College Students

Question

Multiply as indicated. $$\frac{x^{2}-25}{x^{2}-3 x-10} \cdot \frac{x+2}{x}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+5}{x}\) and the domain of the expression is all real numbers except \(x = 0\) and \(x = -2\).

Step 1: Factorization of the First Fraction

The first step is to factorize the first fraction. The numerator \(x^2-25\) is a difference of squares and can be factored as \((x-5)(x+5)\). The denominator \(x^2-3x-10\) is a quadratic equation, it can be factored as \((x-5)(x+2)\). Hence, the first fraction becomes \(\frac{(x-5)(x+5)}{(x-5)(x+2)}\).

Step 2: Simplify the First Fraction by Cancelling Out Common Factors

The numerator and denominator of first fraction share a common factor which is \((x-5)\). So, it cancels out. The first fraction simplifies to \(\frac{x+5}{x+2}\).

Step 3: Simplifying the Expression by Multiplying the Fractions

After dealing with each fraction separately, you should multiply both fractions together. Therefore, \(\frac{x+5}{x+2} * \frac{x+2}{x} = \frac{(x+5)(x+2)}{x(x+2)}\)

Step 4: Cancelling Out Common Factors in Final Expression

The final step is to cancel out the common factors in the numerator and denominator leaving us with final answer which is \(\frac{x+5}{x}\).

Step 5: Discuss the Domain of the Expression

It should also be noted that the expression is undefined when \(x = 0\) and \(x = -2\) because division by zero is not defined, hence the domain of the expression is all real numbers except \(x=0\) and \(x=-2\).