Problem 32 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x^{3}-25 x}{4 x^{2}} \cdot \frac{2 x^{2}-2}{x^{2}-6 x+5} \div \frac{x^{2}+5 x}{7 x+7}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{x}{2} \)

Step 1: Factorize the Numerators and Denominators

The factorized form is given by: \( \frac{(x-5)(x^{2}+5)}{4x(x+1)} \cdot \frac{2x(x-1)}{(x-5)(x-1)} \div \frac{x(x+5)}{7(x+1)} \)

Step 2: Multiply and Divide as indicated

Perform multiplication and division from left to right: \( = \frac{(x-5)(x^{2}+5)(2x(x-1))}{4x(x+1)*(x-5)(x-1)} \div \frac{x(x+5)}{7(x+1)} =\frac{(x-5)(x^{2}+5)(2x(x-1))(7)(x+1)}{4x(x+1)*(x-5)(x-1)x(x+5)} \)

Step 3: Cancel out Common Factors

Cancel out common factors on top and bottom to simplify: \( =\frac{2x^{2}}{4x} = \frac{x}{2} \)