Question

perform the following operation and express in simplest form.\\(
\frac{5x + 25}{x^3 - x^2} \cdot \frac{x^2 - 1}{x^2 + 6x + 5}
\\)
answer
attempt 1 out of 2

Explanation:

Step1: Factor each expression

Factor \(5x + 25\) as \(5(x + 5)\), \(x^3 - x^2\) as \(x^2(x - 1)\), \(x^2 - 1\) as \((x + 1)(x - 1)\) (using difference of squares: \(a^2 - b^2=(a + b)(a - b)\)), and \(x^2 + 6x + 5\) as \((x + 1)(x + 5)\) (factoring quadratic: find two numbers that multiply to 5 and add to 6, which are 1 and 5). So the expression becomes:
\(\frac{5(x + 5)}{x^2(x - 1)} \cdot \frac{(x + 1)(x - 1)}{(x + 1)(x + 5)}\)

Step2: Cancel common factors

Cancel out the common factors \((x + 5)\), \((x - 1)\), and \((x + 1)\) from the numerator and the denominator. After canceling, we have:
\(\frac{5}{x^2}\) (since \(x
eq0, \pm1, - 5\) to avoid division by zero)

Answer:

\(\frac{5}{x^2}\)