Identifying Polynomial Functions With Zeros At X=1 And X=-3
In the realm of mathematics, polynomial functions hold a significant place, serving as the building blocks for various mathematical models and equations. Understanding the behavior of these functions, particularly their zeros, is crucial for solving problems in algebra, calculus, and beyond. In this article, we embark on a journey to decode polynomial functions, focusing on identifying the function that possesses zeros at x=1 and x=-3. We'll delve into the concept of zeros, explore methods for verifying them, and ultimately pinpoint the correct polynomial function from the given options.
Unveiling the Concept of Zeros
At the heart of our exploration lies the concept of zeros, also known as roots or x-intercepts. A zero of a polynomial function is a value of x that makes the function equal to zero. In simpler terms, it's the point where the graph of the polynomial function intersects the x-axis. Zeros play a pivotal role in understanding the behavior of polynomial functions, as they provide insights into the function's solutions, factors, and overall shape.
To illustrate this concept, consider a simple polynomial function like f(x) = x - 2. The zero of this function is x = 2, because when we substitute x = 2 into the function, we get f(2) = 2 - 2 = 0. Graphically, this means the line represented by the function intersects the x-axis at the point (2, 0).
The number of zeros a polynomial function can have is related to its degree, which is the highest power of x in the polynomial. A polynomial of degree n can have at most n zeros, although some zeros may be repeated. For instance, a quadratic function (degree 2) can have up to two zeros, while a cubic function (degree 3) can have up to three zeros.
Methods for Verifying Zeros
Now that we understand the concept of zeros, let's explore methods for verifying whether a given value is indeed a zero of a polynomial function. There are two primary methods we can employ: direct substitution and synthetic division.
1. Direct Substitution
The most straightforward method is direct substitution. We simply substitute the given value of x into the polynomial function and evaluate the expression. If the result is zero, then the value is a zero of the function. If the result is not zero, then the value is not a zero.
For example, let's say we want to verify if x = 1 is a zero of the polynomial function f(x) = x^2 - 1. We substitute x = 1 into the function: f(1) = (1)^2 - 1 = 1 - 1 = 0. Since the result is zero, we can conclude that x = 1 is indeed a zero of the function.
2. Synthetic Division
Synthetic division is a more efficient method for verifying zeros, especially when dealing with polynomials of higher degrees. It's a shorthand method for dividing a polynomial by a linear factor of the form (x - c), where c is the potential zero. If the remainder after synthetic division is zero, then c is a zero of the polynomial.
Let's illustrate synthetic division with an example. Suppose we want to verify if x = -2 is a zero of the polynomial function f(x) = x^3 + 2x^2 - x - 2. We set up the synthetic division as follows:
-2 | 1 2 -1 -2
| -2 0 2
----------------
1 0 -1 0
The last number in the bottom row is the remainder, which is 0 in this case. Since the remainder is zero, we can conclude that x = -2 is a zero of the function.
Pinpointing the Correct Polynomial Function
With the understanding of zeros and methods for verifying them, we can now tackle the task of identifying the polynomial function with zeros at x = 1 and x = -3. We are presented with four options:
A. f(x) = x^3 - 7x + 6 B. f(x) = x^3 + x + 6 C. f(x) = x^3 - 2x^2 - 3x D. f(x) = x^3 - 2x^2 - 7x - 6
To determine the correct function, we will use direct substitution to test whether x = 1 and x = -3 are zeros for each option.
A. f(x) = x^3 - 7x + 6
Let's substitute x = 1: f(1) = (1)^3 - 7(1) + 6 = 1 - 7 + 6 = 0
Now, let's substitute x = -3: f(-3) = (-3)^3 - 7(-3) + 6 = -27 + 21 + 6 = 0
Since both x = 1 and x = -3 result in f(x) = 0, option A is a potential candidate.
B. f(x) = x^3 + x + 6
Substitute x = 1: f(1) = (1)^3 + 1 + 6 = 1 + 1 + 6 = 8 ≠0
Since f(1) is not equal to 0, option B is not the correct function.
C. f(x) = x^3 - 2x^2 - 3x
Substitute x = 1: f(1) = (1)^3 - 2(1)^2 - 3(1) = 1 - 2 - 3 = -4 ≠0
Since f(1) is not equal to 0, option C is not the correct function.
D. f(x) = x^3 - 2x^2 - 7x - 6
Substitute x = 1: f(1) = (1)^3 - 2(1)^2 - 7(1) - 6 = 1 - 2 - 7 - 6 = -14 ≠0
Since f(1) is not equal to 0, option D is not the correct function.
Based on our analysis, only option A, f(x) = x^3 - 7x + 6, satisfies the condition of having zeros at x = 1 and x = -3. Therefore, option A is the correct polynomial function.
Factoring and the Connection to Zeros
Delving deeper into the connection between zeros and polynomial functions, we encounter the concept of factoring. Factoring is the process of expressing a polynomial as a product of simpler polynomials, often linear factors. Each linear factor corresponds to a zero of the polynomial.
For instance, if a polynomial function has a zero at x = c, then (x - c) is a factor of the polynomial. Conversely, if (x - c) is a factor of a polynomial, then x = c is a zero of the polynomial.
In our case, we identified that f(x) = x^3 - 7x + 6 has zeros at x = 1 and x = -3. This means (x - 1) and (x + 3) are factors of the polynomial. To find the complete factorization, we can perform polynomial division or synthetic division to divide f(x) by (x - 1) or (x + 3). Let's divide by (x - 1) using synthetic division:
1 | 1 0 -7 6
| 1 1 -6
------------
1 1 -6 0
The quotient is x^2 + x - 6, which can be further factored as (x + 3)(x - 2). Therefore, the complete factorization of f(x) is (x - 1)(x + 3)(x - 2), and the zeros are x = 1, x = -3, and x = 2.
This connection between zeros and factors provides a powerful tool for analyzing polynomial functions. By finding the zeros, we can determine the factors, and vice versa. This understanding is crucial for solving polynomial equations, graphing polynomial functions, and tackling various mathematical problems.
Real-World Applications of Polynomial Functions
Polynomial functions aren't confined to the theoretical realm of mathematics; they find extensive applications in various real-world scenarios. From modeling physical phenomena to designing engineering systems, polynomial functions serve as indispensable tools for understanding and predicting behavior.
In physics, polynomial functions are used to describe the trajectory of projectiles, the motion of objects under gravity, and the behavior of electrical circuits. For example, the height of a ball thrown into the air can be modeled using a quadratic function, which is a polynomial of degree 2.
In engineering, polynomial functions are used in designing bridges, buildings, and other structures. The shape of a suspension bridge cable, for instance, can be approximated using a polynomial function. Polynomial functions also play a crucial role in control systems, where they are used to model and regulate the behavior of machines and processes.
In economics, polynomial functions are used to model cost curves, revenue curves, and profit functions. These models help businesses make informed decisions about pricing, production levels, and investments.
In computer graphics, polynomial functions are used to create smooth curves and surfaces. Bezier curves, which are widely used in computer-aided design (CAD) and animation software, are based on polynomial functions.
These are just a few examples of the many real-world applications of polynomial functions. Their versatility and ability to model diverse phenomena make them essential tools in various fields of science, engineering, and technology.
Conclusion
In this comprehensive exploration, we have delved into the world of polynomial functions, focusing on the concept of zeros and their significance. We've learned how to verify zeros using direct substitution and synthetic division, and we successfully identified the polynomial function with zeros at x = 1 and x = -3. Furthermore, we explored the connection between zeros and factors, and we discussed the wide-ranging real-world applications of polynomial functions.
Understanding polynomial functions is not just an academic exercise; it's a fundamental skill that empowers us to solve problems, model phenomena, and make informed decisions in various aspects of life. As you continue your mathematical journey, remember the power and versatility of polynomial functions, and embrace the challenges and opportunities they present.
