Question

(3y^2-6y)/(y^2-y-2)

Answer 1

To simplify the expression \((3y^2 - 6y)/(y^2 - y - 2)\), we can start with factoring both the numerator and the denominator.

**Step 1: Factor the numerator**

The numerator is \(3y^2 - 6y\). We can factor out a common term:

\[
3y^2 - 6y = 3y(y - 2)
\]

**Step 2: Factor the denominator**

The denominator is \(y^2 - y - 2\). We need to factor this quadratic. We look for two numbers that multiply to \(-2\) and add to \(-1\). Those numbers are \(-2\) and \(1\). Thus, we can factor:

\[
y^2 - y - 2 = (y - 2)(y + 1)
\]

**Step 3: Rewrite the original expression**

Now, substituting the factored forms back into the expression gives us:

\[
\frac{3y(y - 2)}{(y - 2)(y + 1)}
\]

**Step 4: Cancel common factors**

We can see that \((y - 2)\) is a common factor in both the numerator and the denominator. As long as \(y \neq 2\), we can cancel it out:

\[
\frac{3y \cancel{(y - 2)}}{\cancel{(y - 2)}(y + 1)} = \frac{3y}{y + 1}
\]

**Final Answer:**

The simplified expression is:

\[
\frac{3y}{y + 1}, \quad \text{for } y \neq 2
\]