Finding Real Zeros: G(x) = 4(x^2 + 36)(x - 1)(x + 3)^2

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Hey guys! Today, let's dive into a math problem where we need to find all the real zeros of the function g(x) = 4(x^2 + 36)(x - 1)(x + 3)^2. This might sound a bit intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. Finding the zeros of a function is a fundamental concept in algebra and calculus, and it's super useful in many real-world applications. So, let's get started and see how we can tackle this problem together!

Understanding Zeros of a Function

Before we jump into the specifics of our function, let's quickly recap what zeros actually are. In simple terms, the zeros of a function are the values of x that make the function equal to zero. Graphically, these are the points where the function's graph intersects the x-axis. Finding these points is crucial for understanding the behavior of the function. Zeros help us understand where the function's output is neither positive nor negative, providing critical insights into the function's nature. They play a significant role in various mathematical applications, including solving equations, analyzing graphs, and understanding the roots of polynomials.

For example, if we have a function f(x), and we find that f(2) = 0, then x = 2 is a zero of the function. This means that the graph of f(x) will cross the x-axis at the point (2, 0). Zeros are also known as roots or solutions of the equation f(x) = 0. There are several methods to find the zeros of a function, including factoring, using the quadratic formula, or employing numerical methods for more complex functions. In the case of polynomial functions, the zeros provide essential information about the function's degree and leading coefficient, which in turn helps in sketching the graph of the function. Real-world applications of finding zeros include optimizing processes, modeling physical phenomena, and solving engineering problems.

Breaking Down the Function g(x)

Now, let's take a closer look at our function: g(x) = 4(x^2 + 36)(x - 1)(x + 3)^2. We can see that it's a polynomial function, which is great because polynomial functions have some nice properties that make finding their zeros easier. The function is given in factored form, which is super helpful. Factored form means the function is expressed as a product of several factors, and each factor can potentially give us a zero. By examining the factors, we can identify the values of x that make the entire function equal to zero. This approach simplifies the process of finding zeros compared to dealing with an expanded polynomial. The factored form also allows us to quickly determine the multiplicity of each zero, which indicates how many times a particular zero appears as a root of the equation. This is crucial for sketching the graph of the function, as the multiplicity affects the behavior of the graph at each zero.

Let's break down each part of the function:

  • The constant 4 doesn't affect the zeros, because 4 will never be equal to zero. Constants only scale the function vertically but do not change where the graph crosses the x-axis.
  • The term (x^2 + 36) is interesting. Notice that x^2 is always non-negative for real numbers, and adding 36 to it means this term will always be positive. This term has no real roots because x^2 can never be equal to -36 for any real number x. Therefore, this quadratic factor does not contribute any real zeros to the function. However, it does have complex roots, which we will not consider when looking for real zeros.
  • The term (x - 1) is a linear factor. Setting this equal to zero, we get x - 1 = 0, which gives us a zero at x = 1. This is a straightforward real zero. Linear factors are easy to handle because they directly provide a real zero when set to zero and solved for x.
  • The term (x + 3)^2 is a squared factor. This means we have the factor (x + 3) appearing twice. Setting this equal to zero, we get (x + 3)^2 = 0, which gives us a zero at x = -3. Because it's squared, this zero has a multiplicity of 2, meaning the graph will touch the x-axis at x = -3 but not cross it. Multiplicity plays a crucial role in understanding the graph's behavior near the zeros, as even multiplicities result in the graph touching the x-axis, while odd multiplicities result in the graph crossing the x-axis.

Finding the Real Zeros

Okay, so now we know what we're looking for, let's find those real zeros! Remember, real zeros are the values of x that make g(x) = 0. Since our function is already factored, this makes our job way easier. We just need to set each factor equal to zero and solve for x. The process involves setting each factor of the function to zero, as these factors are the components that, when equal to zero, make the entire function zero. This method leverages the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero. By systematically solving each factor, we ensure that we find all possible zeros of the function. This method is particularly efficient when dealing with polynomial functions that are already in factored form, saving significant time and effort compared to other methods.

  1. Consider the factor (x - 1):

    Set x - 1 = 0. Adding 1 to both sides, we get x = 1. So, one real zero is x = 1. This zero is a simple zero, meaning it has a multiplicity of 1, and the graph of the function will cross the x-axis at this point. The linear nature of the factor (x - 1) makes it straightforward to identify this zero, as it directly corresponds to the x-intercept of the line. Understanding simple zeros is fundamental in graphing functions and solving equations, as they provide key points where the function changes sign.

  2. Consider the factor (x + 3)^2:

    Set (x + 3)^2 = 0. Taking the square root of both sides, we get x + 3 = 0. Subtracting 3 from both sides, we get x = -3. So, another real zero is x = -3. This zero has a multiplicity of 2, meaning the graph touches the x-axis at this point but does not cross it. The squared factor indicates that the function has a repeated root at x = -3. Multiplicity is crucial in analyzing the behavior of the function near the zeros, as it affects the shape of the graph. In this case, the graph will bounce off the x-axis at x = -3 instead of crossing it.

  3. Consider the factor (x^2 + 36):

    Set x^2 + 36 = 0. Subtracting 36 from both sides, we get x^2 = -36. Taking the square root of both sides, we get x = ±√(-36). Since we're looking for real zeros, and we can't take the square root of a negative number and get a real result, this factor gives us no real zeros. However, it does have complex zeros, which are x = 6i and x = -6i, but these are not relevant to our problem of finding real zeros. Complex zeros are important in other contexts, such as understanding the complete solution set of the equation, but they do not correspond to points where the graph intersects the real x-axis.

The Real Zeros of g(x)

Alright, guys, we've done the work! We've analyzed each factor of the function and found the real zeros. So, what are they? The real zeros of the function g(x) = 4(x^2 + 36)(x - 1)(x + 3)^2 are x = 1 and x = -3. Remember that x = -3 has a multiplicity of 2. Summarizing our findings, we have identified the values of x that make the function equal to zero. These zeros are crucial for understanding the behavior and graph of the function. The real zeros represent the points where the graph of the function intersects the x-axis, providing key information about the function's roots and solutions. In this case, the zeros x = 1 and x = -3 are essential for sketching the graph and analyzing the function's characteristics.

Importance of Real Zeros

Finding real zeros isn't just a math exercise; it's super useful in many real-world situations. Real zeros can represent equilibrium points in physical systems, break-even points in business models, or critical values in optimization problems. For example, in physics, zeros might represent points of stability in a system. In economics, they could represent the quantity of goods needed to break even. Understanding real zeros helps in making informed decisions and predictions in various fields. The ability to find and interpret real zeros is a valuable skill in both academic and practical settings, allowing for a deeper understanding of the underlying dynamics of a system or model.

Conclusion

So, there you have it! We've successfully found all the real zeros of the function g(x) = 4(x^2 + 36)(x - 1)(x + 3)^2. Remember, the key is to break down the function into its factors and set each factor equal to zero. Don't forget to consider the multiplicity of the zeros, as it tells us how the graph behaves at those points. By following these steps, you can confidently find the real zeros of any polynomial function. Understanding these concepts is fundamental in mathematics and opens the door to solving more complex problems in various fields. Keep practicing, and you'll become a pro at finding zeros in no time! Awesome job, guys!