Simplify (3x^2+14x+8)/(3x^2-x-2): A Step-by-Step Guide

Hey guys! Let's dive into simplifying rational expressions. These might look a little intimidating at first, but don't worry – we'll break it down step by step. We're going to tackle an expression that looks like this: (3x^2 + 14x + 8) / (3x^2 - x - 2). Sounds fun, right? Our goal is to simplify it completely, and I promise, it's totally doable! We'll focus on factoring the numerator and the denominator separately. Factoring is like reverse multiplication, where we break down a polynomial into smaller expressions that, when multiplied together, give us the original polynomial. This is super important because it allows us to identify common factors that we can cancel out, ultimately simplifying the entire expression. Think of it like reducing a fraction – you find a common factor in the numerator and denominator and divide both by it. It's the same idea here, but with polynomials! Once we've factored both parts, we'll look for those common factors, cancel them, and voilà, we'll have our simplified expression. Remember, practice makes perfect, so don't be afraid to try these out on your own. The more you practice, the easier it'll become to spot those factors and simplify these expressions like a pro!

Factoring the Numerator: 3x² + 14x + 8

Alright, let's start with the top part of our expression, the numerator: 3x² + 14x + 8. To factor this quadratic expression, we need to find two binomials that, when multiplied together, give us this exact polynomial. There are a couple of ways to approach this, but I like to use the "ac method" – it’s a systematic way to break down the problem. Here's how it works: First, identify the coefficients 'a', 'b', and 'c' in our quadratic expression (which is in the standard form ax² + bx + c). In this case, a = 3, b = 14, and c = 8. Next, we multiply 'a' and 'c': 3 * 8 = 24. Now, we need to find two numbers that multiply to 24 and add up to 'b', which is 14. Let's think about the factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Which pair adds up to 14? Bingo! 2 and 12. Now, we rewrite the middle term (14x) using these two numbers: 3x² + 2x + 12x + 8. See what we did there? We split 14x into 2x + 12x. This allows us to factor by grouping. We group the first two terms and the last two terms: (3x² + 2x) + (12x + 8). Now, we factor out the greatest common factor (GCF) from each group. From the first group, we can factor out an 'x', leaving us with x(3x + 2). From the second group, we can factor out a '4', leaving us with 4(3x + 2). Notice something cool? Both groups now have a common factor of (3x + 2). This is exactly what we want! We factor out this common binomial: (3x + 2)(x + 4). And there you have it! We've factored the numerator: 3x² + 14x + 8 = (3x + 2)(x + 4). Factoring quadratics might seem tricky at first, but with practice, you'll get the hang of it. Remember, the key is to break it down step by step, use a systematic method like the 'ac method', and always double-check your work by multiplying the factors back together to make sure you get the original expression. You got this!

Factoring the Denominator: 3x² - x - 2

Okay, now let's tackle the denominator of our rational expression: 3x² - x - 2. We're going to use the same factoring technique we used for the numerator, the "ac method". This consistency will help us stay organized and make the whole process smoother. Remember, the goal is to find two binomials that, when multiplied, give us 3x² - x - 2. So, let's break it down. First, identify the coefficients 'a', 'b', and 'c'. Here, a = 3, b = -1 (notice the negative sign!), and c = -2. Multiply 'a' and 'c': 3 * -2 = -6. Now, we need to find two numbers that multiply to -6 and add up to 'b', which is -1. Let's think about the factors of -6: 1 and -6, -1 and 6, 2 and -3, -2 and 3. Which pair adds up to -1? That's right, 2 and -3. Now, we rewrite the middle term (-x) using these two numbers: 3x² + 2x - 3x - 2. We've split -x into 2x - 3x. This sets us up perfectly for factoring by grouping. Group the first two terms and the last two terms: (3x² + 2x) + (-3x - 2). Factor out the GCF from each group. From the first group, we can factor out an 'x', leaving us with x(3x + 2). From the second group, we can factor out a '-1' (be careful with those negative signs!), leaving us with -1(3x + 2). Notice again that we have a common binomial factor: (3x + 2). This is great news! We factor out this common binomial: (3x + 2)(x - 1). And boom, we've factored the denominator: 3x² - x - 2 = (3x + 2)(x - 1). Just like with the numerator, practice is key. The more you factor, the quicker you'll become at spotting the right numbers and applying the grouping technique. Remember to always double-check your work by multiplying the factors back together. Feeling confident? Awesome! We're one step closer to simplifying the whole expression.

Simplifying the Expression

Okay, guys, we've done the heavy lifting! We've successfully factored both the numerator and the denominator. Now comes the really satisfying part: simplifying the whole expression. This is where we get to cancel out common factors and make things look a lot cleaner. So, let's recap what we've got: Our original expression was (3x² + 14x + 8) / (3x² - x - 2). We factored the numerator and found it to be (3x + 2)(x + 4). We factored the denominator and found it to be (3x + 2)(x - 1). Now, let's put it all together: [(3x + 2)(x + 4)] / [(3x + 2)(x - 1)]. Do you see any factors that appear in both the numerator and the denominator? Yes! We have a (3x + 2) in both places. This means we can cancel them out, just like we would simplify a regular fraction by canceling common factors. So, we cancel out the (3x + 2) from the top and the bottom. What are we left with? (x + 4) / (x - 1). And that, my friends, is our simplified expression! We've taken a seemingly complicated rational expression and reduced it to its simplest form. Isn't that awesome? Now, before we celebrate too much, there's one little thing we need to consider: restrictions. Restrictions are values of 'x' that would make the original denominator equal to zero. Why do we care? Because division by zero is a big no-no in math – it's undefined. So, we need to identify any values of 'x' that would make our original denominator, 3x² - x - 2, equal to zero. We already factored this, so we know it's (3x + 2)(x - 1). This expression will be zero if either (3x + 2) = 0 or (x - 1) = 0. Solving these equations gives us x = -2/3 and x = 1. These are our restrictions. So, our final simplified expression is (x + 4) / (x - 1), with the restrictions x ≠ -2/3 and x ≠ 1. We've simplified the expression and made sure we haven't accidentally introduced any undefined situations. You're doing great! Remember, simplifying rational expressions is all about factoring and canceling common factors. Keep practicing, and you'll become a master at it.

Final Answer

Okay, let's wrap this up and give the final answer in the format the question requested. We successfully simplified the expression (3x² + 14x + 8) / (3x² - x - 2). We factored the numerator to get (3x + 2)(x + 4). We factored the denominator to get (3x + 2)(x - 1). We canceled the common factor of (3x + 2). This left us with the simplified expression (x + 4) / (x - 1). So, to answer the question directly:

• Numerator: x + 4
• Denominator: x - 1

We did it! We took a potentially intimidating rational expression, broke it down into manageable steps, factored, canceled, and simplified. Remember, guys, the key to success with these types of problems is practice. The more you work through them, the more comfortable you'll become with factoring and simplifying. And don't forget about those restrictions! Always check for values that would make the original denominator zero. You've got this! Keep up the awesome work, and happy simplifying!