Finding Polynomial Equation With Given Zeros

When discussing polynomial equations and their solutions, understanding the relationship between zeros and factors is crucial. In this article, we'll explore how to determine a polynomial equation given its zeros. Specifically, we'll address the problem of finding a polynomial p with zeros at x = 1/5, x = -4, and x = 2. This exploration will not only provide a solution to this specific problem but also offer a general method for constructing polynomials from their roots. This article will guide you through the steps to construct such a polynomial, ensuring you grasp the underlying principles and can apply them to various scenarios. Before diving into the solution, let's clarify some key concepts. A zero of a polynomial, also known as a root, is a value of x that makes the polynomial equal to zero. For example, if p(a) = 0, then x = a is a zero of the polynomial p(x). The zeros of a polynomial are intimately connected to its factors. If x = a is a zero of p(x), then (x - a) is a factor of p(x). This relationship forms the basis for constructing polynomials from their zeros. For each zero, we can create a corresponding factor, and by multiplying these factors together, we obtain the polynomial. The degree of the polynomial is determined by the number of zeros (counting multiplicity). A polynomial with n zeros will generally be of degree n. However, the problem often presents multiple possible solutions because multiplying the polynomial by any non-zero constant does not change its zeros. Therefore, we are usually looking for a possible equation rather than the equation. With these concepts in mind, let’s delve into the specifics of finding a polynomial equation with the given zeros.

Understanding Zeros and Factors

In addressing the challenge of constructing a polynomial from its zeros, a foundational understanding of the relationship between zeros and factors is essential. Polynomial zeros are the values of x for which the polynomial evaluates to zero. For instance, if a polynomial p(x) equals zero when x is 1/5, -4, or 2, these values are considered the zeros or roots of the polynomial. Each zero corresponds to a factor of the polynomial. Specifically, if x = a is a zero, then (x - a) is a factor. This principle is a cornerstone of polynomial algebra, allowing us to reverse-engineer the polynomial from its solutions. To illustrate, the zero x = 1/5 corresponds to the factor (x - 1/5). Similarly, the zero x = -4 corresponds to the factor (x + 4), and the zero x = 2 corresponds to the factor (x - 2). By multiplying these factors together, we can construct a polynomial that has the desired zeros. However, it’s important to note that multiplying a polynomial by any non-zero constant does not alter its zeros. This means there are infinitely many polynomials with the same zeros, differing only by a constant factor. The simplest polynomial is obtained by directly multiplying the factors derived from the zeros. In this case, we would multiply (x - 1/5), (x + 4), and (x - 2) together. This will give us a polynomial with the specified zeros. Expanding the product of these factors provides the polynomial in standard form, which is a sum of terms, each consisting of a coefficient and a power of x. The degree of the polynomial (the highest power of x) is equal to the number of zeros, counting multiplicity. In our problem, since we have three distinct zeros (1/5, -4, and 2), the resulting polynomial will be of degree three. Understanding these core principles allows us to approach the problem methodically, ensuring we construct a valid polynomial that satisfies the given conditions. Before we proceed with the construction, it’s helpful to consider how different choices of factors can lead to equivalent polynomials. For instance, we could multiply the entire polynomial by a constant or multiply individual factors by constants, as long as the zeros remain the same. This flexibility is key to recognizing different but correct forms of the solution.

Constructing the Polynomial from Zeros

To construct a polynomial p(x) with zeros at x = 1/5, x = -4, and x = 2, we use the fundamental principle that each zero r corresponds to a factor (x - r). This principle is crucial in polynomial construction and allows us to build the polynomial systematically. First, we identify the factors corresponding to each zero. For the zero x = 1/5, the corresponding factor is (x - 1/5). For the zero x = -4, the corresponding factor is (x - (-4)) which simplifies to (x + 4). And for the zero x = 2, the corresponding factor is (x - 2). Next, we multiply these factors together to form the polynomial. Thus, p(x) can be expressed as p(x) = (x - 1/5)(x + 4)(x - 2). This expression represents a polynomial with the given zeros. To obtain the polynomial in standard form, we would expand this product. However, for the purpose of identifying a possible equation, this factored form is sufficient. We can also multiply the polynomial by any non-zero constant without changing its zeros. For example, we could multiply the entire expression by 5 to eliminate the fraction in the first factor, resulting in p(x) = 5(x - 1/5)(x + 4)(x - 2) = (5x - 1)(x + 4)(x - 2). This is another valid representation of a polynomial with the same zeros. Expanding this further gives us the standard form of the polynomial. We first multiply (5x - 1) by (x + 4), which results in 5x^2 + 20x - x - 4 = 5x^2 + 19x - 4. Then, we multiply this quadratic by (x - 2), giving us: (5x^2 + 19x - 4)(x - 2) = 5x^3 + 19x^2 - 4x - 10x^2 - 38x + 8 = 5x^3 + 9x^2 - 42x + 8. Thus, p(x) = 5x^3 + 9x^2 - 42x + 8 is a polynomial with the specified zeros in standard form. The key takeaway here is the systematic approach of converting zeros to factors and then multiplying these factors to construct the polynomial. This method is applicable to any set of zeros, making it a powerful tool in polynomial algebra. In addition to this, it’s essential to recognize that there are multiple correct answers, differing by a constant multiple. For instance, multiplying the entire polynomial by any non-zero number will not change its zeros, so 2p(x), 3p(x), and so on, are also valid solutions.

Analyzing the Given Options

Now, let's consider the given options and determine which one represents a polynomial with zeros at x = 1/5, x = -4, and x = 2. The options are not provided in the user input, but we can still discuss how to evaluate potential answers. When analyzing options, it’s crucial to check whether each proposed polynomial indeed has the specified zeros. To do this, we substitute each given zero into the polynomial and see if the result is zero. If p(1/5) = 0, p(-4) = 0, and p(2) = 0, then the polynomial has the correct zeros. Another approach is to compare the factors of the proposed polynomial with the factors we derived from the zeros. If the proposed polynomial has factors proportional to (x - 1/5), (x + 4), and (x - 2), then it has the correct zeros. For instance, if one of the options is p(x) = (1/5 x)(-4x)(2x), we can see that this polynomial has zeros at x = 0, x = 0, and x = 0, which are not the zeros we are looking for. This polynomial has a triple root at x = 0, not the zeros x = 1/5, x = -4, and x = 2. On the other hand, if an option looks like a constant multiple of (x - 1/5)(x + 4)(x - 2), it would be a valid solution. For example, p(x) = 5(x - 1/5)(x + 4)(x - 2) is a correct polynomial. The key is to ensure that the zeros of the polynomial match the given zeros. If we encounter a factored form, we can directly compare the factors. If we encounter the polynomial in standard form, we can either attempt to factor it or substitute the zeros to check if they satisfy the equation. In summary, analyzing the given options involves verifying that the proposed polynomials have the correct zeros by either substituting the zeros into the polynomials or comparing their factors with the factors derived from the zeros. This careful verification process is essential to identifying the correct answer.

Conclusion

In conclusion, constructing a polynomial from its zeros involves understanding the fundamental relationship between zeros and factors. Each zero x = r corresponds to a factor (x - r), and multiplying these factors together yields a polynomial with the desired zeros. The polynomial can be multiplied by any non-zero constant without changing its zeros, leading to multiple possible equations. We methodically constructed a polynomial with zeros at x = 1/5, x = -4, and x = 2 by identifying the factors (x - 1/5), (x + 4), and (x - 2), and then multiplying them. We also discussed how to analyze given options by verifying if they have the specified zeros, either by substituting the zeros into the polynomials or comparing their factors. This approach provides a robust method for solving problems involving polynomial zeros and equations, ensuring a solid understanding of polynomial algebra. Understanding these concepts is essential for various mathematical applications, including curve fitting, solving equations, and analyzing functions. The ability to construct polynomials from their zeros is a valuable skill in both theoretical and practical contexts. By mastering these techniques, one can confidently tackle a wide range of problems involving polynomials and their properties. The key is to remember the one-to-one correspondence between zeros and factors and to systematically apply this relationship to construct the desired polynomial.