How To Simplify The Rational Expression (2x^2+19x+35)/(2x^2+11x+15)

Hey guys! Ever stumbled upon a fraction with polynomials and felt a bit lost? Don't worry, we've all been there! This guide will walk you through simplifying rational expressions, making them less intimidating and more manageable. Today, we will dive into simplifying the rational expression: $\frac{2 x^2+19 x+35}{2 x^2+11 x+15}$. By the end of this guide, you’ll be a pro at tackling these problems. So, let's simplify $\frac{2 x^2+19 x+35}{2 x^2+11 x+15}$ together.

What are Rational Expressions?

First off, let's define what we are working with. Rational expressions are essentially fractions where the numerator and denominator are polynomials. Think of them as the algebraic version of regular numerical fractions. Just like you can simplify $\frac{4}{6}$ to $\frac{2}{3}$, you can simplify rational expressions by factoring and canceling out common factors. The key to simplifying rational expressions lies in your ability to factor polynomials. Factoring breaks down a polynomial into its multiplicative components, making it easier to identify common factors between the numerator and the denominator. Strong factoring skills are crucial for simplifying rational expressions, just like understanding multiplication is essential for simplifying numerical fractions. Before we dive into our main problem, let's brush up on factoring. We'll use techniques like recognizing common factors, difference of squares, and factoring quadratic trinomials. This groundwork will make simplifying rational expressions a piece of cake.

Why bother simplifying? Well, simplified expressions are easier to work with in further calculations, like solving equations or graphing functions. Imagine trying to solve an equation with a complex rational expression versus one that's neatly simplified – the latter is much less of a headache! Simplifying also helps in understanding the behavior of the expression, such as its domain and any possible discontinuities. So, let's get started and make these expressions simpler and life easier!

The Problem at Hand: $\frac{2 x^2+19 x+35}{2 x^2+11 x+15}$

Okay, let's jump into our problem: Simplify $\frac{2 x^2+19 x+35}{2 x^2+11 x+15}$. This might look a bit scary at first, but don't sweat it! We'll break it down step by step. The first thing we need to do is factor both the numerator and the denominator. Remember, factoring is like finding the building blocks of a polynomial. We are looking for expressions that, when multiplied together, give us the original polynomial. For the numerator, $2x^2 + 19x + 35$, we need to find two binomials that multiply to give us this quadratic expression. Think of two numbers that multiply to $2 * 35 = 70$ and add up to 19. Those numbers are 14 and 5. So, we can rewrite the middle term and factor by grouping. We rewrite $19x$ as $14x + 5x$, so the numerator becomes $2x^2 + 14x + 5x + 35$. Now, we can factor by grouping. From the first two terms, we can factor out $2x$, giving us $2x(x + 7)$. From the last two terms, we can factor out $5$, giving us $5(x + 7)$. Now we have $2x(x + 7) + 5(x + 7)$, and we can factor out the common factor $(x + 7)$, resulting in $(2x + 5)(x + 7)$.

Now, let's tackle the denominator, $2x^2 + 11x + 15$. We need to do the same thing here: find two binomials that multiply to give us this quadratic. Think of two numbers that multiply to $2 * 15 = 30$ and add up to 11. Those numbers are 6 and 5. So, we rewrite the middle term and factor by grouping. We rewrite $11x$ as $6x + 5x$, so the denominator becomes $2x^2 + 6x + 5x + 15$. Now, we can factor by grouping. From the first two terms, we can factor out $2x$, giving us $2x(x + 3)$. From the last two terms, we can factor out $5$, giving us $5(x + 3)$. Now we have $2x(x + 3) + 5(x + 3)$, and we can factor out the common factor $(x + 3)$, resulting in $(2x + 5)(x + 3)$. So, the original expression can now be written as $\frac{(2x + 5)(x + 7)}{(2x + 5)(x + 3)}$.

Factoring the Numerator and Denominator

Let's zoom in on the factoring process. As we discussed, factoring is the heart of simplifying rational expressions. For the numerator, $2x^2 + 19x + 35$, we used a technique that involves finding two numbers that multiply to the product of the leading coefficient and the constant term (2 * 35 = 70) and add up to the middle coefficient (19). These numbers were 14 and 5. This allowed us to rewrite the middle term, $19x$, as $14x + 5x$, which then enabled us to factor by grouping. Factoring by grouping is a powerful technique where you split the polynomial into pairs of terms and factor out the greatest common factor (GCF) from each pair. In this case, we factored $2x$ from $2x^2 + 14x$ and $5$ from $5x + 35$, leading us to the factored form $(2x + 5)(x + 7)$.

For the denominator, $2x^2 + 11x + 15$, we followed a similar approach. We looked for two numbers that multiply to $2 * 15 = 30$ and add up to 11. These numbers were 6 and 5. We then rewrote $11x$ as $6x + 5x$ and factored by grouping. We factored $2x$ from $2x^2 + 6x$ and $5$ from $5x + 15$, which gave us the factored form $(2x + 5)(x + 3)$. Mastering these factoring techniques is super important for simplifying rational expressions. It's like having the right tools for the job – it makes everything so much easier! Practice these techniques, and you'll find factoring becomes second nature.

Canceling Common Factors

Alright, we've factored both the numerator and the denominator. Now comes the fun part: canceling out common factors! This is where the expression starts to simplify and look much cleaner. We now have the expression $\frac{(2x + 5)(x + 7)}{(2x + 5)(x + 3)}$. Notice anything that appears in both the numerator and the denominator? That's right, we have the common factor $(2x + 5)$. Just like you can cancel out common factors in a regular fraction (like canceling the 2's in $\frac{2 * 3}{2 * 5}$), we can cancel out the $(2x + 5)$ term from both the top and the bottom of our rational expression. Remember, we can only cancel out factors that are multiplied, not terms that are added or subtracted. So, we can cancel out $(2x + 5)$ from the numerator and the denominator, leaving us with $\frac{x + 7}{x + 3}$.

And that's it! We've simplified the rational expression. It's like magic, but it's actually just algebra! The key takeaway here is that canceling common factors is what simplifies the expression. It's like removing the clutter and getting to the core of the expression. But, it's crucial to remember that this canceling is valid only when the factors are identical and present in both the numerator and the denominator. Always double-check your factors before you cancel them out. This step is the essence of simplifying rational expressions, and once you nail it, you're golden!

The Simplified Expression: $\frac{x+7}{x+3}$

So, after all the factoring and canceling, we've arrived at our simplified expression: $\frac{x + 7}{x + 3}$. This is the simplest form of our original rational expression, $\frac{2 x^2+19 x+35}{2 x^2+11 x+15}$. Isn't it satisfying to see how a complex-looking expression can be reduced to something so much cleaner and easier to understand? This simplified form is not only more aesthetically pleasing but also much easier to work with if you were to, say, graph this function or solve an equation involving it. Simplifying rational expressions is all about making things more manageable and revealing the underlying structure of the expression. Now, let's quickly recap the steps we took to get here. First, we factored both the numerator and the denominator. This involved recognizing patterns and using techniques like factoring by grouping. Then, we identified and canceled out the common factors. This step is crucial because it's what actually simplifies the expression. And finally, we were left with our simplified form, $\frac{x + 7}{x + 3}$.

But hold on, there's one more important thing to consider! We need to think about the values of $x$ for which our original expression is undefined. Remember, a rational expression is undefined when the denominator is equal to zero. So, we need to find the values of $x$ that make the original denominator, $2x^2 + 11x + 15$, equal to zero. We already factored this as $(2x + 5)(x + 3)$, so the denominator is zero when $2x + 5 = 0$ or $x + 3 = 0$. This means $x = -\frac{5}{2}$ or $x = -3$. These values are not in the domain of the original expression, and even though we canceled out the $(2x + 5)$ factor, we still need to remember that $x$ cannot be $-\frac{5}{2}$. So, while $\frac{x + 7}{x + 3}$ is the simplified form, it's essential to keep in mind the restrictions on $x$ from the original expression.

Checking the Answer Choices

Now that we've simplified the expression to $\frac{x + 7}{x + 3}$, let's take a look at the answer choices provided in the original problem and see which one matches our result. The answer choices were:

A. $\frac{19 x+35}{11 x+15}$ B. $\frac{7}{3}$ C. $\frac{x+7}{x+3}$ D. $\frac{2 x+7}{2 x+3}$

By comparing our simplified expression with the answer choices, we can clearly see that option C, $\frac{x + 7}{x + 3}$, matches our result. So, C is the correct answer! This step is crucial in problem-solving – always make sure to check your answer against the given options. It's easy to make a small mistake along the way, so verifying your answer can save you from choosing the wrong option. Also, looking at the answer choices can sometimes give you a hint about how to approach the problem. If you see simplified expressions as answer choices, it's a strong indication that you'll need to factor and cancel common factors. Remember, math problems often have clues embedded within them, so keep your eyes peeled!

Tips and Tricks for Simplifying Rational Expressions

Alright, guys, let's wrap things up with some extra tips and tricks that will make simplifying rational expressions even easier! These are the little nuggets of wisdom that can help you tackle these problems with confidence and finesse. First off, always factor completely. This is probably the most crucial tip. Make sure you've factored the numerator and denominator as much as possible before you start canceling anything out. Sometimes, there might be hidden common factors that you'll miss if you don't factor completely. Think of it like cleaning your room – you need to take everything out before you can start organizing it properly. Similarly, factor everything before you start simplifying.

Next up, watch out for the difference of squares. This is a common pattern that can make factoring much easier. If you see something like $a^2 - b^2$, remember that it factors into $(a + b)(a - b)$. Spotting this pattern can save you a lot of time and effort. Another handy trick is to look for a greatest common factor (GCF) first. If there's a common factor in all the terms of a polynomial, factor it out right away. This will make the remaining polynomial simpler to factor. For example, if you have $4x^2 + 8x + 12$, you can factor out a 4 first, giving you $4(x^2 + 2x + 3)$.

And lastly, always check for restrictions on the variable. As we discussed earlier, a rational expression is undefined when the denominator is zero. So, make sure to identify any values of $x$ that would make the denominator zero and exclude them from your solution. This is a crucial step to ensure your answer is complete and accurate. So, there you have it – some extra tips and tricks to help you simplify rational expressions like a pro. Keep these in mind, practice regularly, and you'll be simplifying expressions in your sleep!

Practice Problems

To really nail this skill, practice is key! So, let’s go through a few extra practice problems. These will give you the chance to apply what you’ve learned and build your confidence. Remember, the more you practice, the more comfortable you'll become with factoring and simplifying. Here’s our first practice problem: Simplify $\frac{x^2 - 4}{x^2 + 4x + 4}$. Take a moment to try this one on your own before we walk through the solution.

Alright, let's break this one down. The first step is to factor both the numerator and the denominator. The numerator, $x^2 - 4$, is a difference of squares, so it factors into $(x + 2)(x - 2)$. The denominator, $x^2 + 4x + 4$, is a perfect square trinomial, which factors into $(x + 2)(x + 2)$ or $(x + 2)^2$. So, our expression becomes $\frac{(x + 2)(x - 2)}{(x + 2)(x + 2)}$. Now we can cancel out the common factor of $(x + 2)$, leaving us with $\frac{x - 2}{x + 2}$. And that's the simplified form! Remember to also consider the restriction that $x$ cannot be $-2$, as that would make the original denominator zero.

Let's try another one: Simplify $\frac{3x^2 + 9x}{x^2 + 6x + 9}$. Again, pause for a moment and try it on your own first. For this one, we can start by factoring out a GCF from the numerator. The greatest common factor of $3x^2$ and $9x$ is $3x$, so we can factor that out to get $3x(x + 3)$. The denominator, $x^2 + 6x + 9$, is another perfect square trinomial, which factors into $(x + 3)(x + 3)$ or $(x + 3)^2$. So, our expression becomes $\frac{3x(x + 3)}{(x + 3)(x + 3)}$. We can cancel out one factor of $(x + 3)$ from the numerator and the denominator, leaving us with $\frac{3x}{x + 3}$. And there you have it – another rational expression simplified! Don’t forget, $x$ cannot be $-3$ in this case.

Keep practicing these types of problems, and you’ll become a master of simplifying rational expressions in no time! Practice makes perfect, and with each problem you solve, you'll build more confidence and skill.

Conclusion

So, guys, we've reached the end of our journey into simplifying rational expressions! We've covered a lot of ground, from understanding what rational expressions are to factoring polynomials, canceling common factors, and identifying restrictions on the variable. You've learned that simplifying these expressions is all about breaking them down into their simplest form, making them easier to work with and understand. Remember, the key steps are to factor the numerator and denominator completely, cancel out any common factors, and always consider the restrictions on the variable. Factoring is your superpower here, so make sure you're comfortable with different factoring techniques like factoring by grouping, difference of squares, and recognizing perfect square trinomials.

We tackled the problem $\frac{2 x^2+19 x+35}{2 x^2+11 x+15}$ and successfully simplified it to $\frac{x + 7}{x + 3}$. We also walked through several practice problems, giving you the chance to apply your newfound skills. Simplifying rational expressions might seem challenging at first, but with practice and the right techniques, you can conquer any expression that comes your way. So, keep practicing, stay curious, and don't be afraid to ask for help when you need it. You've got this!

Now go out there and simplify some rational expressions! You've got the tools, the knowledge, and the confidence to tackle them head-on. Happy simplifying!