A Polynomial Function Has Roots -5 And 1. Which Function Could Represent This Polynomial?

When we delve into the world of polynomial functions, understanding the relationship between roots and the function's equation is crucial. The roots of a polynomial function are the values of x for which the function equals zero. In simpler terms, these are the points where the graph of the function intersects the x-axis. This article aims to provide a comprehensive explanation of how to determine a polynomial function given its roots, focusing on the specific case where the roots are -5 and 1. We will explore the underlying principles, discuss various representations of polynomial functions, and analyze the given options to identify the correct function. By the end of this exploration, you will have a solid grasp of how roots define a polynomial and how to translate this knowledge into practical problem-solving.

The Fundamental Concept: Roots and Factors

The cornerstone of understanding this problem lies in the relationship between the roots of a polynomial and its factors. A root, denoted as r, of a polynomial function f(x) implies that f(r) = 0. This fundamental concept allows us to construct the factors of the polynomial. If r is a root, then (x - r) is a factor of the polynomial. This is because when x = r, the factor (x - r) becomes zero, making the entire polynomial zero.

Let's apply this to our specific scenario. We are given that the roots of the polynomial function are -5 and 1. This means that f(-5) = 0 and f(1) = 0. Applying the principle mentioned above, we can deduce the factors of the polynomial. For the root -5, the corresponding factor is (x - (-5)), which simplifies to (x + 5). Similarly, for the root 1, the corresponding factor is (x - 1). Therefore, the polynomial function must have factors of (x + 5) and (x - 1).

This understanding is crucial because it allows us to build the polynomial function from its roots. By identifying the factors, we can construct a general form of the polynomial. In this case, the polynomial function can be represented as f(x) = a(x + 5)(x - 1), where a is a non-zero constant. The constant a accounts for the possibility of the polynomial being scaled vertically. For instance, both (x + 5)(x - 1) and 2(x + 5)(x - 1) have the same roots, but they represent different parabolas if we were to graph them.

The concept of roots and factors is not just limited to quadratic polynomials (polynomials of degree 2). It extends to polynomials of any degree. For example, if a polynomial has roots r1, r2, r3, ..., rn, then the polynomial can be expressed in the form f(x) = a(x - r1)(x - r2)(x - r3)...(x - rn). This general form highlights the significance of roots in defining the polynomial function.

Understanding this fundamental relationship between roots and factors is essential for solving a wide range of polynomial problems. It forms the basis for many algebraic techniques, including polynomial factorization, solving polynomial equations, and graphing polynomial functions. In our specific problem, this principle directly leads us to identifying the correct form of the polynomial function.

Analyzing the Given Options

Now that we have established the fundamental relationship between roots and factors, let's analyze the given options in light of this understanding. We know that the roots are -5 and 1, and the corresponding factors must be (x + 5) and (x - 1). A polynomial function with these roots can be written in the form f(x) = a(x + 5)(x - 1), where a is a constant.

Let's examine each option:

A. f(x) = (x + 5)(x + 1)

This option has factors (x + 5) and (x + 1). The factor (x + 5) corresponds to the root -5, which is correct. However, the factor (x + 1) corresponds to the root -1, which is incorrect. We need a factor that corresponds to the root 1, not -1. Therefore, this option is not the correct representation of the polynomial function.

B. f(x) = (x - 5)(x - 1)

This option has factors (x - 5) and (x - 1). The factor (x - 5) corresponds to the root 5, which is incorrect. Our roots are -5 and 1, not 5 and 1. The factor (x - 1) correctly corresponds to the root 1, but the presence of (x - 5) disqualifies this option as the correct function.

C. f(x) = (x - 5)(x + 1)

This option has factors (x - 5) and (x + 1). Similar to option B, the factor (x - 5) corresponds to the root 5, which is incorrect. The factor (x + 1) corresponds to the root -1, which is also incorrect. Neither of these factors aligns with our given roots of -5 and 1, making this option incorrect.

D. f(x) = (x + 5)(x - 1)

This option has factors (x + 5) and (x - 1). The factor (x + 5) corresponds to the root -5, which is correct. The factor (x - 1) corresponds to the root 1, which is also correct. This option perfectly matches our derived form of the polynomial function, f(x) = a(x + 5)(x - 1), with a = 1. Therefore, this is the correct representation of the polynomial function.

Through this detailed analysis, we can confidently conclude that option D is the only option that accurately represents a polynomial function with roots -5 and 1. This methodical approach of examining each option based on the fundamental principles of roots and factors is essential for solving similar problems.

Expanding the Polynomial and Understanding its Form

To further solidify our understanding, let's expand the correct polynomial function and explore its standard form. We identified that f(x) = (x + 5)(x - 1) is the correct representation. Expanding this product gives us:

f(x) = (x + 5)(x - 1) = x(x - 1) + 5(x - 1) = x^2 - x + 5x - 5 = x^2 + 4x - 5

Thus, the polynomial function in standard form is f(x) = x^2 + 4x - 5. This is a quadratic polynomial, which is characterized by its highest power of x being 2. The standard form of a quadratic polynomial is f(x) = ax^2 + bx + c, where a, b, and c are constants. In our case, a = 1, b = 4, and c = -5.

The standard form provides valuable information about the polynomial. The coefficient a determines the direction of the parabola (the graph of a quadratic function). If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. In our case, a = 1, which means the parabola opens upwards. The constant c represents the y-intercept of the parabola. In our case, c = -5, which means the parabola intersects the y-axis at the point (0, -5).

Furthermore, the roots of the polynomial can be found by setting f(x) = 0 and solving for x. We already know the roots are -5 and 1. Let's verify this by using the quadratic formula, which is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in our values, we get:

x = (-4 ± √(4^2 - 4(1)(-5))) / (2(1)) = (-4 ± √(16 + 20)) / 2 = (-4 ± √36) / 2 = (-4 ± 6) / 2

This gives us two solutions:

x = (-4 + 6) / 2 = 2 / 2 = 1

x = (-4 - 6) / 2 = -10 / 2 = -5

These solutions confirm our given roots of -5 and 1. This process demonstrates how the roots, factors, and standard form of a polynomial are interconnected. Understanding these relationships is essential for a comprehensive understanding of polynomial functions.

Generalizing to Higher-Degree Polynomials

While we have focused on a quadratic polynomial in this example, the principles we discussed can be generalized to polynomials of higher degrees. The fundamental relationship between roots and factors remains the same: if r is a root of a polynomial f(x), then (x - r) is a factor of f(x). This principle holds true regardless of the degree of the polynomial.

For instance, consider a cubic polynomial (a polynomial of degree 3). If a cubic polynomial has roots r1, r2, and r3, then it can be written in the form f(x) = a(x - r1)(x - r2)(x - r3), where a is a constant. Similarly, for a quartic polynomial (a polynomial of degree 4) with roots r1, r2, r3, and r4, the polynomial can be written as f(x) = a(x - r1)(x - r2)(x - r3)(x - r4), and so on.

The degree of the polynomial indicates the maximum number of roots it can have. A polynomial of degree n can have at most n roots. However, these roots may not all be distinct. A root can have a multiplicity greater than 1, meaning it appears multiple times as a root. For example, the polynomial f(x) = (x - 2)^2 has a root of 2 with a multiplicity of 2.

Understanding the relationship between the degree of a polynomial and the number of roots is crucial for solving polynomial equations and analyzing polynomial functions. It allows us to predict the general shape of the graph of the polynomial and identify key features such as intercepts and turning points.

In conclusion, the problem of finding a polynomial function given its roots highlights the fundamental connection between roots and factors. By understanding this relationship, we can construct polynomial functions from their roots and analyze their properties. This knowledge is essential for a deeper understanding of algebra and calculus, as polynomial functions are foundational concepts in these areas of mathematics. Through careful analysis and application of these principles, we can confidently solve a wide range of polynomial problems.

Conclusion

In summary, to identify a polynomial function given its roots, the key is to understand the relationship between roots and factors. If a polynomial has roots -5 and 1, it must have factors of (x + 5) and (x - 1). Therefore, the correct representation of the polynomial function is f(x) = (x + 5)(x - 1). This understanding forms the basis for more advanced concepts in polynomial algebra and calculus.