Solving For Zeros: Understanding Rational Functions

Hey everyone! Today, we're diving into the world of rational functions and figuring out how to find where they equal zero. This is super important stuff in math, and we'll break it down step by step so you can ace it! Let's get started, guys!

Understanding Rational Functions and Their Zeros

Okay, so first things first: What exactly is a rational function? Well, it's simply a function that's expressed as a fraction, where both the numerator (the top part) and the denominator (the bottom part) are polynomials. Think of it like this: r(x) = p(x) / q(x), where p(x) and q(x) are your polynomial buddies. Now, the zeros of a rational function are the x-values that make the whole function equal to zero. But here's the kicker: A rational function can only be zero when its numerator is zero. Why? Because if the top part of the fraction is zero, the whole fraction becomes zero (as long as the bottom part isn't also zero, but we'll get to that in a bit). So, the trick to finding the zeros of a rational function is to focus on the numerator, set it equal to zero, and solve for x. Easy peasy, right?

Let's consider the given rational function: r(x) = (x^3 - 4x + 3) / (x^4 + 2x - 4). Our goal is to find the values of x for which r(x) = 0. As we've learned, the value of x makes the rational function equals to zero only when the numerator x^3 - 4x + 3 equals zero, so we need to solve the equation x^3 - 4x + 3 = 0. Solving this is our main task here, and this is where some cool math tricks come into play. We are not going to focus on the denominator x^4 + 2x - 4 because for the rational function equals zero, the denominator has to be not equal to zero. The process involves identifying the zeros of the numerator. When the numerator of a rational function is zero, the entire function equals zero, provided the denominator is not simultaneously zero at the same x-value. We're effectively simplifying the problem by focusing on the numerator. This is a common and crucial strategy when dealing with rational functions, as it isolates the points where the function crosses the x-axis, providing key insights into its behavior and characteristics. These x-values will be the zeros of the rational function. This is critical for understanding the function's behavior, where the graph of the function will intersect the x-axis, where the function's value is precisely zero.

Focusing on the Numerator and Solving for x

So, we need to solve the cubic equation x^3 - 4x + 3 = 0. Cubic equations, unlike the linear and quadratic ones, can be a bit more challenging to solve directly. There isn't a simple, universal formula like the quadratic formula. But don't worry; we have some tools to help us out. The first trick in our toolbox is looking for integer roots using the Rational Root Theorem. This theorem tells us that if there are any rational roots (roots that can be expressed as fractions), they must be factors of the constant term (in our case, 3) divided by factors of the leading coefficient (in our case, 1). Therefore, we need to test out some possible roots. We can try out the factors of 3, which are ±1 and ±3. If we plug in x = 1 into the equation, we get 1^3 - 4(1) + 3 = 0. Hey, it works! This means that x = 1 is a root of the equation. This is a major win because it means we can factor the cubic expression. Once we find one root, then we can use polynomial division or synthetic division to find the other roots easily. Now, we can use synthetic division to divide the cubic polynomial by (x - 1). When we do that, we get a quadratic equation as our quotient. Then, we can solve this quadratic equation.

Applying Synthetic Division

Let's apply synthetic division. Dividing x^3 - 4x + 3 by (x - 1), we find that the result is x^2 + x - 3. Now our cubic equation becomes (x - 1)(x^2 + x - 3) = 0. So, one of the roots is obviously x = 1 (from the factor (x - 1)). But wait, we still have a quadratic equation to solve: x^2 + x - 3 = 0. Now, we can use the quadratic formula to solve for the other two roots. The quadratic formula is your best friend when it comes to solving equations of this form. Remember it? It is: x = (-b ± √(b^2 - 4ac)) / 2a. Here, a = 1, b = 1, and c = -3. Plugging in those values, we get: x = (-1 ± √(1^2 - 4 * 1 * -3)) / (2 * 1), which simplifies to x = (-1 ± √13) / 2. So, we get two more potential roots: x = (-1 + √13) / 2 and x = (-1 - √13) / 2. Let's calculate these values, we have: x ≈ 1.303 and x ≈ -2.303. Those are the zeros of the quadratic equation, which means they are also the zeros of the rational function. Thus the values of x for which r(x) = 0 are 1, -2.303, and 1.303.

Identifying the Correct Answer Choice

Based on our calculations, the values of x that make r(x) = 0 are approximately -2.303, 1.000, and 1.303. Let's look at the answer choices provided:

• (A) x = -2.303 and x = 1.000 only: This option is incorrect because it misses the root 1.303.
• (B) x = -1.643 and x = 1.144 only: This option is incorrect as it contains values that aren't the solution to r(x) = 0.
• (C) x = -2.303, x = 1.000, and x = 1.303 only: This is our winner! This option correctly identifies all three roots.
• (D) x = -2.303, x = -1.643, and x = 1.000 only: This option is incorrect since it has an extra root -1.643 and excludes the root 1.303.

Therefore, the correct answer is (C).

Conclusion: Mastering Rational Function Zeros

Great job, everyone! We've successfully navigated the process of finding the zeros of a rational function. Remember, the key is to focus on the numerator, solve for x, and always double-check your work. Keep practicing these problems, and you'll become a pro in no time! Keep in mind, when faced with rational functions, always remember that the zeros occur where the numerator equals zero, assuming the denominator isn't simultaneously zero at the same x-value. Using a combination of algebraic manipulation, the rational root theorem, synthetic division, and the quadratic formula, we have found that the zeros of our given rational function are approximately -2.303, 1, and 1.303. These x-values represent the points at which the function's graph intersects the x-axis, providing critical information about its behavior. By understanding these concepts and practicing different types of problems, you'll become more and more proficient in tackling any problem that involves rational functions, and this is crucial for advanced math and in real-world applications as well. That is all for this time. See you later, and keep practicing!