Simplify Rational Expression: X^2+5x+6 / X^2-x-6

Hey guys! Today, we're diving deep into the world of rational expressions. If you've ever looked at something like (x^2+5x+6)/(x^2-x-6) and felt a bit intimidated, don't sweat it! We're going to break it down, simplify it, and figure out which of the options is actually equal to it when x isn't -2 or 3. This is a super common type of problem in algebra, and once you get the hang of factoring, it becomes way less scary. So, grab your notebooks, and let's get this done!

Understanding Rational Expressions and Why We Simplify Them

First off, what exactly is a rational expression? Think of it like a fraction, but instead of just numbers, you've got algebraic expressions (that's where the x and x^2 come in) in the numerator and the denominator. So, (x^2+5x+6)/(x^2-x-6) is a perfect example. Now, why do we even bother simplifying these bad boys? Well, just like simplifying a regular fraction (say, 4/8 becomes 1/2), simplifying a rational expression makes it easier to work with. It helps us see the core relationship between the numerator and the denominator, identify restrictions, and solve equations more efficiently. The key to simplifying these expressions lies in factoring both the numerator and the denominator. Once factored, we can cancel out any common factors, provided those factors don't equal zero. That's where the conditions like x != -2 or 3 come into play – they tell us the values of x that would make a denominator zero, which is a big no-no in math (division by zero is undefined, remember?). So, our main mission today is to factor the top and bottom of our given expression and then see what cancels out.

Factoring the Numerator: x^2 + 5x + 6

Alright, let's tackle the numerator first: x^2 + 5x + 6. We need to find two numbers that multiply to give us 6 (the constant term) and add up to give us 5 (the coefficient of the x term). Think about the factors of 6: they can be 1 and 6, or 2 and 3. Which pair adds up to 5? Yep, it's 2 and 3! So, we can rewrite the numerator as (x + 2)(x + 3). See? Not so bad! This is called factoring a quadratic trinomial. We're essentially reversing the FOIL method (First, Outer, Inner, Last) that you'd use to multiply two binomials. If you were to multiply (x + 2)(x + 3) back out, you'd get x*x (first) + x*3 (outer) + 2*x (inner) + 2*3 (last), which simplifies to x^2 + 3x + 2x + 6, and then x^2 + 5x + 6. Perfect match!

Factoring the Denominator: x^2 - x - 6

Now, let's move on to the denominator: x^2 - x - 6. This one's a little trickier because of the negative sign. We need two numbers that multiply to give us -6 and add up to give us -1 (remember, if there's no number written before x, the coefficient is 1, and here it's negative, so -1). Let's list the factors of -6:

• 1 and -6 (add up to -5)
• -1 and 6 (add up to 5)
• 2 and -3 (add up to -1)
• -2 and 3 (add up to 1)

Bingo! The pair 2 and -3 works because 2 * -3 = -6 and 2 + (-3) = -1. So, we can factor the denominator as (x + 2)(x - 3). Again, you can check this by multiplying it back: (x + 2)(x - 3) = x*x + x*(-3) + 2*x + 2*(-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6. Nailed it!

Simplifying the Expression: Canceling Common Factors

We've done the hard part – factoring! Now, let's put it all together. Our original expression is:

$$\frac{x^2+5 x+6}{x^2-x-6}$$

After factoring, we have:

$$\frac{(x+2)(x+3)}{(x+2)(x-3)}$$

Look closely, guys. Do you see any common factors in the numerator and the denominator? Yes, there it is: (x + 2)! Since we're given that x != -2, we know that (x + 2) is not equal to zero. This is crucial because we can only cancel out factors that are not zero. So, we can cancel out the (x + 2) from the top and the bottom.

$$\frac{\cancel{(x+2)}(x+3)}{\cancel{(x+2)}(x-3)}$$

What's left? We're left with:

$$\frac{x+3}{x-3}$$

This is our simplified expression!

Checking the Conditions and Final Answer

Remember those conditions we were given at the start? x != -2 or 3. Let's see why they're important.

• The original denominator is x^2 - x - 6. If we set this to zero, x^2 - x - 6 = 0, and we factor it, we get (x + 2)(x - 3) = 0. This equation is true if x + 2 = 0 (which means x = -2) or if x - 3 = 0 (which means x = 3). So, the original expression is undefined when x = -2 or x = 3.
• Our simplified expression is (x + 3)/(x - 3). The denominator here is x - 3. This expression is undefined only when x - 3 = 0, which means x = 3.

Notice that our simplified expression is defined at x = -2 (it would be (-2+3)/(-2-3) = 1/-5), but the original expression is not. This is why it's important to state the restrictions from the original expression even after simplifying. The simplified expression is equal to the original expression only for the values of x where the original expression was defined. So, our simplified form (x + 3)/(x - 3) is equivalent to the original expression given that x != -2 and x != 3.

Now, let's look at the options provided:

A. (x+3)/(x-3) B. (x+2)/(x-2) C. (x+2)/(x-3)

Our simplified expression is (x+3)/(x-3), which perfectly matches option A. So, the correct answer is A!

Conclusion: Mastering Rational Expressions

And there you have it, folks! We successfully simplified the rational expression by factoring the numerator and the denominator and canceling out the common factor (x + 2). Remember, the key steps are always:

1. Factor the numerator.
2. Factor the denominator.
3. Identify any common factors.
4. Cancel out the common factors, making sure they are not equal to zero (use the given restrictions).

By following these steps, you can tackle any rational expression problem thrown your way. Keep practicing, and you'll become a simplification pro in no time! If you ever get stuck, just break it down piece by piece, focus on the factoring, and you'll be golden. Happy math-ing!