Polynomial Equation From Zeros With Multiplicity How To Find It

Hey guys! Today, we're diving into the fascinating world of polynomials and how to construct their equations when we know their zeros (or roots). This is a fundamental concept in algebra, and it's super useful in various mathematical applications. So, let's break it down step by step and make sure we understand it thoroughly. We'll tackle a specific example where we're given the zeros and their multiplicities and asked to find the correct polynomial equation from a set of choices.

Understanding Zeros and Multiplicity

Before we jump into the problem, let's quickly recap what zeros and multiplicities mean in the context of polynomials. The zeros of a polynomial, also known as roots, are the values of x that make the polynomial equal to zero. In other words, these are the points where the graph of the polynomial intersects the x-axis. For example, if x = a is a zero of a polynomial f(x), then f(a) = 0. The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. If a zero has a multiplicity of n, it means the corresponding factor (x - zero) appears n times in the factored form of the polynomial. This affects the behavior of the graph at that zero. For instance, a zero with multiplicity 1 crosses the x-axis, a zero with multiplicity 2 touches the x-axis and turns around (it's a turning point), and so on. The multiplicity plays a crucial role in determining the shape and characteristics of the polynomial function's graph, and understanding it is key to solving polynomial-related problems. In essence, when dealing with polynomials, think of zeros as the foundational points that define the polynomial's behavior, and multiplicity as the guiding factor that shapes its graph around those points. Grasping these concepts will make polynomial manipulations and problem-solving significantly easier and more intuitive.

Problem Statement

Alright, let's get to the problem at hand. We are tasked with finding the equation of a polynomial given the following information:

• Zero: $\frac{2}{5}$ with multiplicity 2
• Zero: $-\frac{2}{5}$ with multiplicity 1

And we have four options to choose from:

A. $f(x)=125 x^3-50 x^2-20 x+8$ B. $f(x)=25 x^3-50 x^2-4 x+8$ C. $f(x)=25 x^2-4$ D. $f(x)=125 x^3+50 x^2-20 x-8$

Our mission, should we choose to accept it (and we do!), is to figure out which of these equations represents the polynomial with the specified zeros and multiplicities. We'll do this by constructing the polynomial from its factored form, using the given zeros and their multiplicities, and then comparing our result with the options provided. This approach ensures that we're building the polynomial from the ground up, guaranteeing accuracy and a solid understanding of the process. So, buckle up, because we're about to embark on a polynomial-solving adventure!

Constructing the Polynomial

Okay, let's put on our constructor hats and build this polynomial from scratch! We know that if a number r is a zero of a polynomial, then (x - r) is a factor of that polynomial. Also, the multiplicity of the zero tells us how many times that factor appears. So, given that $\frac{2}{5}$ is a zero with multiplicity 2, we know that $\left(x - \frac{2}{5}\right)^2$ is a factor. Similarly, since $-\frac{2}{5}$ is a zero with multiplicity 1, we have $\left(x + \frac{2}{5}\right)$ as a factor. To get the polynomial, we multiply these factors together:

$$f(x) = \left(x - \frac{2}{5}\right)^2 \left(x + \frac{2}{5}\right)$$

Now, let's expand this expression. First, we'll expand the squared term:

$$\left(x - \frac{2}{5}\right)^2 = x^2 - 2\left(x\right)\left(\frac{2}{5}\right) + \left(\frac{2}{5}\right)^2 = x^2 - \frac{4}{5}x + \frac{4}{25}$$

Next, we multiply this by the remaining factor $\left(x + \frac{2}{5}\right)$:

$$f(x) = \left(x^2 - \frac{4}{5}x + \frac{4}{25}\right) \left(x + \frac{2}{5}\right)$$

Now, we distribute each term:

$$f(x) = x^2\left(x + \frac{2}{5}\right) - \frac{4}{5}x\left(x + \frac{2}{5}\right) + \frac{4}{25}\left(x + \frac{2}{5}\right)$$

$$f(x) = x^3 + \frac{2}{5}x^2 - \frac{4}{5}x^2 - \frac{8}{25}x + \frac{4}{25}x + \frac{8}{125}$$

Combine like terms:

$$f(x) = x^3 + \left(\frac{2}{5} - \frac{4}{5}\right)x^2 + \left(-\frac{8}{25} + \frac{4}{25}\right)x + \frac{8}{125}$$

$$f(x) = x^3 - \frac{2}{5}x^2 - \frac{4}{25}x + \frac{8}{125}$$

To get rid of the fractions, we can multiply the entire polynomial by 125 (the least common multiple of the denominators):

$$125f(x) = 125\left(x^3 - \frac{2}{5}x^2 - \frac{4}{25}x + \frac{8}{125}\right)$$

$$125f(x) = 125x^3 - 50x^2 - 20x + 8$$

So, we've constructed our polynomial! Now, let's compare it to the given options.

Comparing with Options

Alright, we've built our polynomial, and it looks like this:

$$125f(x) = 125x^3 - 50x^2 - 20x + 8$$

Now, let's line up our contenders:

A. $f(x)=125 x^3-50 x^2-20 x+8$ B. $f(x)=25 x^3-50 x^2-4 x+8$ C. $f(x)=25 x^2-4$ D. $f(x)=125 x^3+50 x^2-20 x-8$

By carefully comparing our constructed polynomial with the options, we can clearly see that option A matches perfectly. Option A, $f(x) = 125x^3 - 50x^2 - 20x + 8$, is exactly what we derived. The coefficients and signs align, confirming that this is the correct polynomial equation for the given zeros and multiplicities. The other options have different coefficients or signs, making them incorrect. Therefore, through our step-by-step construction and comparison, we've confidently pinpointed the right answer. So, give yourselves a pat on the back, polynomial problem-solvers! We've nailed it!

Final Answer

Therefore, the correct answer is:

A. $f(x)=125 x^3-50 x^2-20 x+8$

Great job, everyone! We successfully found the equation of the polynomial by understanding the relationship between zeros, multiplicities, and the factored form of a polynomial. Keep practicing, and you'll become polynomial pros in no time!