Simplifying Algebraic Expression Worksheets - Free Printable

We are asked to simplify the following rational expression:

$$
\frac{v^2 - 8v - 9}{v^2 + 3v + 2}
$$

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Step 1: Factor the numerator and denominator

#### Factor the numerator: $ v^2 - 8v - 9 $

We need two numbers that multiply to $-9$ and add to $-8$.
Let's find such a pair:

- $1$ and $-9$: $1 \cdot (-9) = -9$, $1 + (-9) = -8$ ✔

So:
$$
v^2 - 8v - 9 = (v + 1)(v - 9)
$$

#### Factor the denominator: $ v^2 + 3v + 2 $

We need two numbers that multiply to $2$ and add to $3$.
Try:
- $1$ and $2$: $1 \cdot 2 = 2$, $1 + 2 = 3$ ✔

So:
$$
v^2 + 3v + 2 = (v + 1)(v + 2)
$$

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Step 2: Rewrite the expression with factored forms

$$
\frac{(v + 1)(v - 9)}{(v + 1)(v + 2)}
$$

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Step 3: Cancel common factors

Both numerator and denominator have a factor of $ (v + 1) $, so we can cancel it as long as $ v \neq -1 $ (since division by zero is undefined).

After canceling:
$$
\frac{v - 9}{v + 2}, \quad \text{for } v \neq -1
$$

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✔ Final Answer:

$$
\boxed{\frac{v - 9}{v + 2}}, \quad v \neq -1
$$

This is the simplified form of the given ratio.