Zeros Of Polynomial Function: F(x) = X^4 - 30x^2 + 125

Let's dive into finding the zeros of the polynomial function f(x) = x^4 - 30x^2 + 125 and figuring out the multiplicity of each zero. This might sound a bit intimidating, but don't worry, we'll break it down step by step so it's super easy to follow. Finding zeros is a fundamental skill in algebra and calculus, and understanding multiplicity helps us grasp the behavior of the function around those zeros.

Understanding Zeros and Multiplicity

Before we jump into the problem, let's make sure we're all on the same page about what zeros and multiplicity actually mean. Guys, zeros of a function are simply the values of x that make the function equal to zero. In other words, they are the x-values where the graph of the function intersects the x-axis. These are also sometimes called roots or solutions of the equation f(x) = 0.

Now, multiplicity refers to the number of times a particular zero appears as a root of the polynomial equation. For example, if (x - a) appears n times as a factor in the factored form of the polynomial, then a is a zero with multiplicity n. The multiplicity tells us something important about how the graph behaves near that zero. If the multiplicity is odd (like 1, 3, 5, etc.), the graph crosses the x-axis at that zero. If the multiplicity is even (like 2, 4, 6, etc.), the graph touches the x-axis at that zero but doesn't cross it—it just bounces off. This behavior is crucial for sketching polynomial functions accurately.

Consider a simple example: f(x) = (x - 2)^3 (x + 1)^2. Here, x = 2 is a zero with multiplicity 3, and x = -1 is a zero with multiplicity 2. Near x = 2, the graph will cross the x-axis, whereas near x = -1, it will bounce off the x-axis.

So, armed with this knowledge, let's tackle the polynomial function f(x) = x^4 - 30x^2 + 125. Our goal is to find all the values of x that make this function equal to zero and determine how many times each of those values appears as a root.

Solving for the Zeros of f(x) = x^4 - 30x^2 + 125

Okay, let's get our hands dirty with the polynomial f(x) = x^4 - 30x^2 + 125. The first thing we want to do is set the function equal to zero: x^4 - 30x^2 + 125 = 0. Now, this looks like a quartic equation, which can be a bit scary, but don't worry! We can use a clever trick to turn it into a quadratic equation, which we know how to solve.

Notice that we have x^4 and x^2 terms. This suggests that we can make a substitution. Let's let y = x^2. Then, y^2 = (x²⁾2 = x^4. Substituting these into our equation gives us:

y^2 - 30y + 125 = 0

Ah, much better! This is a quadratic equation in terms of y. Now we can try to factor this quadratic. We are looking for two numbers that multiply to 125 and add up to -30. Those numbers are -25 and -5. So, we can factor the quadratic as follows:

(y - 25)(y - 5) = 0

This gives us two possible values for y: y = 25 and y = 5. But remember, we want to find the values of x, not y. So, we need to substitute back x^2 for y.

For y = 25, we have x^2 = 25. Taking the square root of both sides gives us x = ±5. So, we have two zeros: x = 5 and x = -5.

For y = 5, we have x^2 = 5. Taking the square root of both sides gives us x = ±√5. So, we have two more zeros: x = √5 and x = -√5.

Therefore, the zeros of the polynomial function f(x) = x^4 - 30x^2 + 125 are x = 5, x = -5, x = √5, and x = -√5.

Determining the Multiplicity of Each Zero

Now that we've found the zeros, the next step is to determine the multiplicity of each zero. To do this, we need to express the polynomial in its factored form. We found that x = 5, x = -5, x = √5, and x = -√5 are the zeros. This means the factors of the polynomial are (x - 5), (x + 5), (x - √5), and (x + √5). Thus, we can write the polynomial as:

f(x) = (x - 5)(x + 5)(x - √5)(x + √5)

Notice that each factor appears only once. This means that each zero has a multiplicity of 1. So:

• x = 5 has a multiplicity of 1.
• x = -5 has a multiplicity of 1.
• x = √5 has a multiplicity of 1.
• x = -√5 has a multiplicity of 1.

Since all the multiplicities are odd, the graph of the function will cross the x-axis at each of these zeros. Understanding the zeros and their multiplicities helps us to sketch the graph of the polynomial accurately and predict its behavior. This is incredibly useful in many applications of polynomial functions.

Summarizing the Results

Alright, let's wrap everything up. We started with the polynomial function f(x) = x^4 - 30x^2 + 125 and we wanted to find its zeros and the multiplicity of each zero. Here’s what we found:

• Zeros: The zeros of the polynomial function are x = 5, x = -5, x = √5, and x = -√5.
• Multiplicity: Each of these zeros has a multiplicity of 1.

To find these zeros, we used a substitution method to transform the quartic equation into a quadratic equation, which we then factored. We found the values of y and then substituted back to find the corresponding values of x. Finally, by writing the polynomial in factored form, we determined that each zero has a multiplicity of 1.

Understanding zeros and their multiplicities is super important in polynomial functions. It helps us to sketch the graph of the function, determine its behavior around the x-axis, and solve various problems involving polynomial equations. So, keep practicing these steps, and you'll become a pro at finding zeros and multiplicities in no time!

Practical Applications and Further Exploration

Knowing how to find the zeros and their multiplicities isn't just an abstract mathematical exercise; it has tons of practical applications in various fields. For instance, in engineering, polynomials are used to model curves and surfaces, and finding their zeros can help determine critical points, such as maximum and minimum values, which are essential for design and optimization. In physics, polynomial functions can describe trajectories and oscillations, and understanding their zeros can help predict system behavior.

In computer graphics, polynomials are used to create smooth curves and surfaces. Being able to find the zeros and understand their multiplicity allows developers to control the shape and behavior of these curves. Furthermore, in signal processing, polynomials are used in filter design, and their zeros play a crucial role in determining the filter's frequency response.

For further exploration, consider investigating how the zeros and their multiplicities relate to the derivatives of the polynomial. For example, a zero with multiplicity greater than 1 will also be a zero of the derivative of the polynomial. This connection is fundamental in calculus and provides additional insights into the behavior of polynomial functions. You can also explore different methods for finding zeros, such as numerical methods like the Newton-Raphson method, which are particularly useful for polynomials with no simple algebraic solutions. Also, look into the relationship between the coefficients of the polynomial and its zeros through Vieta's formulas. Understanding these relationships can provide deeper insights into the structure and behavior of polynomials.

So, guys, keep exploring, keep questioning, and keep applying these concepts to real-world problems. The more you practice and delve into the fascinating world of polynomials, the more you'll appreciate their power and versatility!