Simplify (4x²-13x+3)/(4x²+3x-1): A Step-by-Step Guide

by Kenji Nakamura 54 views

Hey guys! Today, we're diving into simplifying rational expressions, which might sound intimidating, but trust me, it's like solving a puzzle. We'll tackle the expression (4x² - 13x + 3) / (4x² + 3x - 1) step-by-step, making sure we understand each move. So, grab your pencils, and let's get started!

Understanding Rational Expressions

Before we jump into the simplification, let's break down what a rational expression actually is. Simply put, a rational expression is just a fraction where the numerator and the denominator are polynomials. Polynomials, remember, are expressions made up of variables and coefficients, like our 4x² - 13x + 3 and 4x² + 3x - 1. Simplifying these expressions involves factoring, canceling out common factors, and ultimately making the expression as clean and concise as possible.

Think of it like reducing a regular fraction, like 6/8. We find the greatest common factor (GCF), which is 2, and divide both the numerator and the denominator by it, giving us 3/4. We're going to do something very similar with our polynomials, but instead of numbers, we'll be working with algebraic expressions. This is a crucial skill in algebra because it helps in solving equations, understanding function behavior, and even in calculus later on. So, mastering this now will set you up for success in more advanced math courses.

When dealing with rational expressions, there are a few key things to keep in mind. First, always look for common factors in both the numerator and the denominator. This is the golden rule of simplification. Second, remember that you can only cancel out factors, not terms. A factor is something that is multiplied, while a term is something that is added or subtracted. This distinction is super important to avoid making mistakes. Lastly, it's a good practice to state any restrictions on the variable. For example, if the denominator has a variable, we need to make sure that the denominator doesn't equal zero, as division by zero is undefined. These restrictions will become clearer as we work through the problem.

Step 1: Factoring the Numerator (4x² - 13x + 3)

The first part of our simplification journey is to factor the numerator, which is 4x² - 13x + 3. Factoring is like reverse multiplication; we're trying to find two binomials that, when multiplied together, give us our original quadratic expression. There are several methods for factoring, but a common one is the AC method or factoring by grouping. This method is particularly useful when the coefficient of the x² term (in this case, 4) is not 1.

Here’s how the AC method works for our numerator:

  1. Multiply A and C: Multiply the coefficient of the x² term (A = 4) by the constant term (C = 3). So, 4 * 3 = 12.
  2. Find Two Numbers: We need to find two numbers that multiply to 12 and add up to the coefficient of the x term (B = -13). After a little thought, we find that -12 and -1 fit the bill since (-12) * (-1) = 12 and (-12) + (-1) = -13.
  3. Rewrite the Middle Term: Rewrite the middle term (-13x) using the two numbers we found. So, -13x becomes -12x - x. Our expression now looks like this: 4x² - 12x - x + 3.
  4. Factor by Grouping: Group the first two terms and the last two terms: (4x² - 12x) + (-x + 3). Now, factor out the greatest common factor (GCF) from each group. From the first group, we can factor out 4x, and from the second group, we can factor out -1. This gives us: 4x(x - 3) - 1(x - 3).
  5. Final Factorization: Notice that both terms now have a common factor of (x - 3). Factor this out, and we get our final factored form: (4x - 1)(x - 3).

So, we've successfully factored the numerator! This factored form, (4x - 1)(x - 3), is equivalent to our original numerator, 4x² - 13x + 3, but it's in a form that will allow us to simplify our rational expression. Remember, the key to factoring is practice, so don't get discouraged if it seems tricky at first. Keep working at it, and you'll become a factoring pro in no time!

Step 2: Factoring the Denominator (4x² + 3x - 1)

Now that we've conquered the numerator, it's time to tackle the denominator: 4x² + 3x - 1. We'll use the same AC method we used before, which is a reliable way to factor quadratic expressions, especially when the leading coefficient isn't 1. This method allows us to systematically break down the expression into its factors.

Let's walk through the steps:

  1. Multiply A and C: Again, we start by multiplying the coefficient of the x² term (A = 4) by the constant term (C = -1). This gives us 4 * -1 = -4.
  2. Find Two Numbers: Next, we need to find two numbers that multiply to -4 and add up to the coefficient of the x term (B = 3). After considering the factors of -4, we find that 4 and -1 fit perfectly since 4 * -1 = -4 and 4 + (-1) = 3.
  3. Rewrite the Middle Term: Just like before, we rewrite the middle term (3x) using the two numbers we found. So, 3x becomes 4x - x. Our expression now looks like this: 4x² + 4x - x - 1.
  4. Factor by Grouping: We group the first two terms and the last two terms: (4x² + 4x) + (-x - 1). Now, we factor out the greatest common factor (GCF) from each group. From the first group, we can factor out 4x, and from the second group, we can factor out -1. This gives us: 4x(x + 1) - 1(x + 1).
  5. Final Factorization: We notice that both terms now share a common factor of (x + 1). Factoring this out, we arrive at our final factored form: (4x - 1)(x + 1).

So, we've successfully factored the denominator! Our factored form, (4x - 1)(x + 1), is equivalent to the original denominator, 4x² + 3x - 1. With both the numerator and denominator factored, we're now in a prime position to simplify our rational expression. Remember, practice makes perfect, and mastering these factoring techniques will be incredibly useful in your mathematical journey.

Step 3: Simplifying the Expression

Now for the exciting part – simplifying the entire expression! We’ve factored both the numerator and the denominator, and now we're going to put those factored forms together and see what we can cancel out. This is where all our hard work pays off and we get to see the expression in its simplest form. Remember, the goal is to identify common factors that appear in both the numerator and the denominator, and then divide them out.

Let's recap what we've got:

  • Factored Numerator: (4x - 1)(x - 3)
  • Factored Denominator: (4x - 1)(x + 1)

So, our rational expression now looks like this:

(4x - 1)(x - 3) / (4x - 1)(x + 1)

Do you see any common factors? Bingo! Both the numerator and the denominator have the factor (4x - 1). This is fantastic because we can now cancel out this common factor. Think of it like dividing both the top and bottom of a regular fraction by the same number – it doesn't change the value of the fraction, but it does simplify it.

Canceling out the (4x - 1) factor, we’re left with:

(x - 3) / (x + 1)

And there you have it! We've successfully simplified our rational expression. This is the simplest form of the original expression. There are no more common factors to cancel out, and we've made our expression as clean and concise as possible. This simplified form is much easier to work with in further calculations or when graphing.

It's important to remember that simplifying rational expressions is a fundamental skill in algebra. It not only helps you to make expressions easier to work with but also gives you a deeper understanding of the underlying structure of algebraic expressions. Plus, this skill is essential for more advanced topics like calculus, where simplifying complex expressions is a daily task.

Step 4: State Restrictions (Important!)

Before we wrap things up, there's one crucial step we absolutely cannot skip: stating the restrictions. This is a critical part of simplifying rational expressions, and it ensures that our simplified expression is equivalent to the original for all valid values of x. Restrictions come into play because we need to avoid any values of x that would make the denominator of the original expression equal to zero. Remember, division by zero is undefined in mathematics, so we have to exclude those values.

Let's go back to our original denominator, which was 4x² + 3x - 1, or its factored form, (4x - 1)(x + 1). To find the restrictions, we need to set each factor in the original denominator equal to zero and solve for x:

  1. 4x - 1 = 0

    • Add 1 to both sides: 4x = 1
    • Divide both sides by 4: x = 1/4

  2. x + 1 = 0

    • Subtract 1 from both sides: x = -1

So, we have two values of x that would make the original denominator zero: x = 1/4 and x = -1. These are our restrictions. This means that our simplified expression, (x - 3) / (x + 1), is equivalent to the original expression (4x² - 13x + 3) / (4x² + 3x - 1) for all values of x except x = 1/4 and x = -1.

We typically state the restrictions like this: x ≠ 1/4 and x ≠ -1. This notation tells us that x can be any real number except for 1/4 and -1. Including these restrictions is crucial for the completeness and accuracy of our solution. When you're simplifying rational expressions, always remember to check for and state the restrictions.

Why are restrictions so important? Well, imagine we didn't state them. We might mistakenly plug x = 1/4 into our simplified expression and get a perfectly valid numerical answer. However, if we plugged x = 1/4 into the original expression, we'd be dividing by zero, which is a big no-no. Stating the restrictions makes it clear that our simplified expression is only valid for certain values of x, maintaining mathematical integrity.

Final Answer

Alright, guys! We've reached the finish line. After all the factoring, canceling, and stating restrictions, let's present our final answer in a clear and concise way. This is how we would typically present the solution in a math class or on a test.

The simplified form of the rational expression (4x² - 13x + 3) / (4x² + 3x - 1) is:

(x - 3) / (x + 1)

And the restrictions are:

x ≠ 1/4 and x ≠ -1

This complete answer tells us both the simplified expression and the values of x for which that simplification is valid. It's like giving the full story – not just the punchline, but also the context that makes it make sense.

In summary, simplifying rational expressions involves several key steps:

  1. Factoring the Numerator: Breaking down the top part of the fraction into its factors.
  2. Factoring the Denominator: Breaking down the bottom part of the fraction into its factors.
  3. Simplifying the Expression: Canceling out any common factors between the numerator and the denominator.
  4. Stating the Restrictions: Identifying any values of x that would make the original denominator equal to zero and excluding them.

By following these steps, you can confidently simplify rational expressions and tackle more complex algebraic problems. Remember, math is like building with blocks – each concept builds on the previous one. Mastering these foundational skills will set you up for success in all your future math endeavors. Keep practicing, and you'll become a math whiz in no time!