Problem 65 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. $$\left(\frac{y-2}{y^{2}-9 y+18} \cdot \frac{y^{2}-4 y-12}{y+2}\right) \div \frac{y^{2}-4}{y^{2}+5 y+6}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{(y-6)(y^2 + 5y + 6)}{(y-2)(y-3)}\).

Step 1: Factor the Expressions

Begin by factoring the expressions. Using the quadratic factoring approach, we get: \[ \left(\frac{(y-6)(y+3)}{(y-6)(y-3)} \cdot \frac{(y-6)(y+2)}{(y+2)}\right) \div \frac{(y-2)(y+2)}{(y+3)(y+2)} \] After factoring, cancel out the common factors.

Step 2: Perform the Multiplication and Division Operations

Dividing by a fraction is the same as multiplying by its reciprocal. Rearrange and rewrite the expression to get: \[ \left(\frac{(y-6)(y+3)}{(y-3)}\right) \cdot \left(\frac{(y+3)(y+2)}{(y-2)}\right) \]

Step 3: Simplify the Result

Perform the multiplication and simplify the expression to get the final result. \[ \frac{(y-6)(y^2 + 5y + 6)}{(y-2)(y-3)} \]