Simplify (x+2)/(4x²+5x+1)⋅(4x+1)/(x²-4)

by Rajiv Sharma 40 views

Hey guys! Today, we're diving deep into the world of rational expressions and tackling a common challenge: simplifying them. Rational expressions might sound intimidating, but trust me, they're totally manageable once you break them down. We're going to walk through a specific example step-by-step, so you'll not only get the answer but also understand the how and why behind it. So, grab your pencils, and let's get started!

The Problem: A Rational Expression Puzzle

So, the problem we're going to solve today is this: What is the simplest form of the expression: (x+2) / (4x² + 5x + 1) ⋅ (4x+1) / (x²-4)

This looks like a bit of a puzzle, right? We've got fractions with variables, polynomials, and multiplication all mixed in. But don't worry, we'll break it down into smaller, easier-to-handle pieces. Our goal is to simplify this expression as much as possible. We'll achieve this by using key algebraic techniques, like factoring polynomials and canceling common factors. By the end of this guide, you will be a rational expression simplification pro.

Step 1: Factor Everything!

The golden rule when simplifying rational expressions? Factor, factor, factor! Factoring is like taking apart a machine to see its individual components. It allows us to identify common pieces that we can later simplify.

Factoring the Numerators

Let's start with the numerators (the top parts) of our fractions:

  • (x + 2): This guy is already in its simplest form. We can't factor it any further. It's a linear expression, and there are no common factors to pull out. So, we just leave it as is.
  • (4x + 1): Same story here! This is also a linear expression, and it's already in its simplest factored form. There's no greatest common factor other than 1 to extract. So, we're good to go.

Factoring the Denominators

Now, let's move on to the denominators (the bottom parts), which are a bit more interesting:

  • (4x² + 5x + 1): This is a quadratic expression, and quadratics are usually factorable. We need to find two binomials that multiply together to give us this trinomial. There are several techniques to do this. You can use the AC method, trial and error, or factoring by grouping. Let's go through factoring by grouping here.
    • First, multiply the coefficient of the x² term (4) by the constant term (1), which gives us 4.
    • Now, we need to find two numbers that multiply to 4 and add up to the coefficient of the x term (5). Those numbers are 4 and 1.
    • Rewrite the middle term (5x) as the sum of these two numbers (4x + 1x): 4x² + 4x + 1x + 1
    • Now, factor by grouping. Group the first two terms and the last two terms: (4x² + 4x) + (1x + 1)
    • Factor out the greatest common factor (GCF) from each group: 4x(x + 1) + 1(x + 1)
    • Notice that we now have a common binomial factor (x + 1). Factor this out: (x + 1)(4x + 1)
    • So, the factored form of 4x² + 5x + 1 is (x + 1)(4x + 1)
  • (x² - 4): This looks like a difference of squares! Remember the formula: a² - b² = (a + b)(a - b). In our case, a = x and b = 2. So, we can factor this as:
    • (x² - 4) = (x + 2)(x - 2)

Putting It All Together

Now that we've factored everything, let's rewrite our original expression with the factored forms:

(x + 2) / [(x + 1)(4x + 1)] ⋅ (4x + 1) / [(x + 2)(x - 2)]

See how much clearer things are now? We've transformed a seemingly complex expression into a product of simpler factors. This sets us up perfectly for the next step: simplification!

Step 2: Cancel Common Factors

This is the fun part! Now that we've factored everything, we can start canceling out common factors. Think of it like simplifying fractions with numbers – if you have the same factor in the numerator and denominator, you can cancel them out.

Let's look at our factored expression again:

(x + 2) / [(x + 1)(4x + 1)] ⋅ (4x + 1) / [(x + 2)(x - 2)]

We can see some factors that appear in both the numerator and the denominator:

  • (x + 2): We have (x + 2) in the numerator of the first fraction and in the denominator of the second fraction. So, we can cancel these out.
  • (4x + 1): We have (4x + 1) in the denominator of the first fraction and in the numerator of the second fraction. These can also be canceled.

After canceling these common factors, our expression looks like this:

1 / (x + 1) ⋅ 1 / (x - 2)

Notice how much simpler things have become? By canceling common factors, we've significantly reduced the complexity of the expression.

Step 3: Multiply and Simplify

We're almost there! The last step is to multiply the remaining fractions together and see if we can simplify further.

We have:

1 / (x + 1) ⋅ 1 / (x - 2)

To multiply fractions, we simply multiply the numerators together and the denominators together:

(1 ⋅ 1) / [(x + 1)(x - 2)]

This simplifies to:

1 / [(x + 1)(x - 2)]

Now, let's take a look at the denominator. We have (x + 1)(x - 2). We could multiply this out, but in this case, it's generally preferred to leave it in factored form because it's considered the simplest form.

So, our final simplified expression is:

1 / [(x + 1)(x - 2)]

The Answer and Why It Matters

Therefore, the simplest form of the given expression is:

A. 1 / [(x + 1)(x - 2)]

Yay! We did it! We successfully navigated through factoring, canceling, and simplifying to arrive at the final answer.

But simplifying rational expressions isn't just about getting the right answer. It's a fundamental skill in algebra and calculus. Simplified expressions are:

  • Easier to work with: They make further calculations and problem-solving much simpler.
  • Easier to understand: The simplified form often reveals the underlying structure and behavior of the expression.
  • Essential for graphing: Simplified forms help in identifying key features of a function's graph, like asymptotes and intercepts.

So, mastering the art of simplifying rational expressions is a valuable investment in your mathematical journey.

Key Takeaways: Mastering Rational Expressions

Before we wrap up, let's quickly recap the key steps we took to simplify our rational expression. Remember these, and you'll be well-equipped to tackle similar problems:

  1. Factor Everything: This is the most crucial step. Factor all numerators and denominators completely. Look for common factoring patterns like the difference of squares, or use techniques like factoring by grouping.
  2. Cancel Common Factors: Once everything is factored, identify and cancel any factors that appear in both the numerator and denominator. This is where the magic happens, as it significantly simplifies the expression.
  3. Multiply and Simplify: Multiply the remaining numerators and denominators. Double-check if there are any further simplifications possible. In many cases, leaving the denominator in factored form is the preferred way to represent the answer.

Practice Makes Perfect: Level Up Your Skills

Simplifying rational expressions is a skill that gets better with practice. Don't be discouraged if it feels challenging at first. The more you practice, the more comfortable you'll become with the different factoring techniques and simplification strategies.

So, go ahead and find some more problems to work on. Challenge yourself with different types of expressions and factoring patterns. The more you practice, the more confident you'll become in your ability to simplify rational expressions like a pro!

If you have any questions or want to explore more examples, feel free to reach out. Keep up the great work, and happy simplifying!