Problem 28 - College Algebra

Question

Multiply or divide as indicated. $$\frac{x^{2}+x}{x^{2}-4} \div \frac{x^{2}-1}{x^{2}+5 x+6}$$

Step-by-Step Solution

Answer

The answer is \(x(x+3)/(x-2)(x-1)\)

Step 1: Factor the Expressions

Factorize each of the four parts as follows: \(x^{2}+x=x(x+1)\), \(x^{2}-4= (x-2)(x+2)\), \(x^{2}-1 = (x-1)(x+1)\), \(x^{2}+5x+6 = (x+2)(x+3)\)

Step 2: Rewrite the Division as Multiplication

Convert the division of fractions into multiplication of the first fraction and the reciprocal of the second fraction: So, the expression changes to \(\frac{x(x+1)}{(x-2)(x+2)} \times \frac{(x+2)(x+3)}{(x-1)(x+1)}\).

Step 3: Cancel Common Factors

Now, cancel common factors that appear in both the numerator and the denominator. Here (x+2) and (x+1) are common factors: After cancelling, we have \(\frac{x}{x-2} \times \frac{x+3}{x-1}\).

Step 4: Simplify the Result

After cancelling out, proceed by multiplying the fractions normally. The final result is \(\frac{x(x+3)}{(x-2)(x-1)}\).