Polynomial Equation Find With Given Zeros

by ADMIN 42 views

In mathematics, a polynomial is an expression consisting of variables and coefficients, combined using only the operations of addition, subtraction, and multiplication, and non-negative integer exponents. Finding the equation of a polynomial given its zeros is a fundamental concept in algebra. This article will delve into the process of constructing a polynomial equation when provided with its zeros and their respective multiplicities. We will explore the underlying principles, demonstrate the step-by-step procedure, and illustrate the concept with a detailed example. Understanding this process is crucial for various mathematical applications, including curve fitting, function analysis, and solving algebraic equations.

Understanding Polynomial Zeros and Multiplicities

In this section, we will explore the core concepts of polynomial zeros and multiplicities, which form the foundation for constructing polynomial equations. Understanding these concepts is essential for accurately determining the equation of a polynomial given its roots and their behavior.

Polynomial Zeros: Unveiling the Roots

Polynomial zeros, also known as roots, are the values of the variable (usually x) that make the polynomial equal to zero. In simpler terms, they are the points where the graph of the polynomial intersects the x-axis. For instance, if a polynomial f(x) has a zero at x = a, then f(a) = 0. These zeros provide crucial information about the structure and behavior of the polynomial. Each zero corresponds to a factor of the polynomial, which is a linear expression of the form (x - a), where a is the zero. The zeros of a polynomial are fundamental to understanding its behavior and can be used to solve polynomial equations, factor polynomials, and graph polynomial functions.

Multiplicity: Decoding the Zero's Behavior

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. It essentially tells us how many times a factor (x - a) is repeated in the factored form of the polynomial. For example, if a zero a has a multiplicity of 2, it means the factor (x - a) appears twice, i.e., (x - a)². The multiplicity of a zero significantly impacts the graph of the polynomial at that point. If the multiplicity is odd, the graph crosses the x-axis at the zero. If the multiplicity is even, the graph touches the x-axis but does not cross it, creating a turning point. Understanding the multiplicity of zeros allows us to accurately sketch the graph of a polynomial and predict its behavior near its roots. For example, a zero with multiplicity 1 is a simple root where the graph crosses the x-axis, while a zero with multiplicity 2 indicates a turning point where the graph touches the x-axis and bounces back. Higher multiplicities lead to flatter curves near the zero.

In summary, the zeros of a polynomial indicate where the graph intersects the x-axis, and the multiplicity of each zero describes the behavior of the graph at that point. Combining this information is essential for constructing the polynomial equation, as each zero and its multiplicity contribute to the factored form of the polynomial. By understanding these concepts, we can effectively reverse-engineer the polynomial equation from its roots.

Constructing the Polynomial Equation: A Step-by-Step Guide

Now that we understand the significance of zeros and multiplicities, let's outline the systematic process of constructing the polynomial equation. This process involves translating the given zeros and their multiplicities into factors and then combining these factors to form the polynomial. The following steps provide a clear roadmap for this construction:

  1. Identify the Zeros and Their Multiplicities: Begin by carefully listing all the given zeros of the polynomial. For each zero, note its multiplicity, which indicates how many times it appears as a root. This information is the foundation for building the polynomial equation. For example, if you are given zeros of 2 (with multiplicity 1) and -3 (with multiplicity 2), you have the basic building blocks for your polynomial.

  2. Form the Factors: For each zero a, create a corresponding factor in the form (x - a). If a zero has a multiplicity greater than 1, raise the factor to the power of its multiplicity. This step translates the zeros into the algebraic components of the polynomial. For instance, if the zeros are 2 (multiplicity 1) and -3 (multiplicity 2), the factors would be (x - 2) and (x + 3)². The multiplicity determines the exponent of the factor, which in turn affects the shape of the graph at that zero.

  3. Multiply the Factors: Multiply all the factors together to obtain the polynomial equation. This step combines the individual factors into a complete polynomial expression. Expanding the product of the factors will give you the polynomial in its standard form, where terms are arranged in descending order of their exponents. For example, multiplying (x - 2) and (x + 3)² involves first expanding (x + 3)² to get x² + 6x + 9, and then multiplying this quadratic by (x - 2). This process gives the polynomial in its expanded form, which reveals the coefficients and degree of the polynomial.

  4. Adjust the Leading Coefficient (if necessary): If additional information is provided, such as a specific point that the polynomial passes through, you may need to adjust the leading coefficient. This adjustment ensures that the polynomial satisfies the given condition. The leading coefficient is the number that multiplies the highest power of x in the polynomial. Adjusting it involves setting up an equation using the given point and solving for the leading coefficient. For example, if you know the polynomial passes through the point (1, 4), you can substitute x = 1 and f(x) = 4 into the polynomial equation and solve for the leading coefficient. This final step ensures that the polynomial not only has the correct zeros and multiplicities but also fits any additional constraints provided.

By following these steps, you can systematically construct the equation of a polynomial from its zeros and their multiplicities. This process is a fundamental skill in algebra and is essential for solving various mathematical problems involving polynomials.

Example: Constructing a Polynomial Equation

Let's illustrate the process with a concrete example. Suppose we are tasked with finding the equation of a polynomial that has a zero at $ rac{2}{5}$ with multiplicity 2 and another zero at $- rac{2}{5}$ with multiplicity 1. We will follow the steps outlined earlier to construct the polynomial equation.

Step 1: Identify the Zeros and Their Multiplicities

We are given the following zeros and multiplicities:

  • Zero: $ rac{2}{5}$, Multiplicity: 2
  • Zero: $- rac{2}{5}$, Multiplicity: 1

This means that $ rac{2}{5}$ is a repeated root, appearing twice, and $- rac{2}{5}$ appears once as a root.

Step 2: Form the Factors

For each zero, we create a corresponding factor:

  • For the zero $ rac{2}{5}$ with multiplicity 2, the factor is $(x - rac{2}{5})^2$. This factor is squared because the multiplicity is 2, indicating that the root appears twice.
  • For the zero $- rac{2}{5}$ with multiplicity 1, the factor is $(x + rac{2}{5})$. Since the multiplicity is 1, the factor appears as is.

Step 3: Multiply the Factors

Now, we multiply the factors together to form the polynomial equation:

f(x) = (x - rac{2}{5})^2 (x + rac{2}{5})

To simplify, we first expand the squared term:

(x - rac{2}{5})^2 = (x - rac{2}{5})(x - rac{2}{5}) = x^2 - rac{4}{5}x + rac{4}{25}

Next, we multiply this quadratic by the remaining factor:

f(x) = (x^2 - rac{4}{5}x + rac{4}{25})(x + rac{2}{5})

Expanding this product, we get:

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

Combine like terms:

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

Step 4: Adjust the Leading Coefficient (if necessary)

To eliminate fractions and obtain a polynomial with integer coefficients, we can multiply the entire polynomial by the least common multiple (LCM) of the denominators, which is 125:

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

Distribute the 125:

f(x)=125x3−50x2−20x+8f(x) = 125x^3 - 50x^2 - 20x + 8

Therefore, the equation of the polynomial is:

f(x)=125x3−50x2−20x+8f(x) = 125x^3 - 50x^2 - 20x + 8

This polynomial has the specified zeros and multiplicities. The process of constructing the polynomial equation involves understanding the relationship between zeros, factors, and multiplicities, and systematically combining them to form the polynomial expression.

Analyzing the Answer Choices

Now, let's examine the answer choices provided and determine which one matches our derived polynomial equation:

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

Comparing these options with our result, $f(x) = 125x^3 - 50x^2 - 20x + 8$, we can see that option B is the correct answer. The other options do not match the polynomial we constructed based on the given zeros and multiplicities.

Option A has incorrect coefficients for the cubic and linear terms. Option C is a quadratic polynomial, which would only account for the zero at $ rac{2}{5}$ with multiplicity 2, but not the zero at $-\frac{2}{5}$. Option D is a linear polynomial and does not account for any of the given zeros or their multiplicities.

Therefore, option B is the only polynomial that accurately reflects the given zeros and their respective multiplicities.

Conclusion

In conclusion, finding the equation of a polynomial given its zeros and multiplicities is a crucial skill in algebra. This article has provided a comprehensive guide, including a step-by-step process and a detailed example, to help you master this concept. Understanding the relationship between zeros, factors, and multiplicities is key to constructing the polynomial equation accurately. By following the outlined steps and practicing with various examples, you can confidently tackle problems involving polynomial construction. This skill is not only essential for academic success but also for numerous applications in mathematics, engineering, and other scientific fields. Remember, the zeros indicate where the graph intersects the x-axis, and multiplicities describe the behavior of the graph at those intersections. Combining this information allows you to build the polynomial equation effectively. Always double-check your work and ensure that the final polynomial equation satisfies the given conditions.