Verifying Zeros Of Polynomial P(x) = 3x² - 1
In the realm of mathematics, polynomials hold a position of paramount importance, serving as fundamental building blocks for numerous mathematical models and equations. Among the various aspects of polynomial analysis, identifying the zeros of a polynomial holds particular significance. Zeros, also known as roots, are the values of the variable that render the polynomial equal to zero. In this comprehensive article, we embark on a detailed exploration of the polynomial p(x) = 3x² - 1, delving into the process of verifying whether specific values, namely x = -1/√3 and x = 2/√3, are indeed zeros of this polynomial. This meticulous examination will not only solidify our understanding of polynomial zeros but also provide valuable insights into the broader concepts of polynomial evaluation and root finding.
Understanding Polynomial Zeros
Before we delve into the specifics of our example polynomial, let's take a moment to solidify our understanding of polynomial zeros. A zero of a polynomial p(x) is a value of x that satisfies the equation p(x) = 0. In simpler terms, it's the value of x that makes the polynomial expression equal to zero. These zeros hold immense significance as they represent the points where the graph of the polynomial intersects the x-axis. The zeros of a polynomial provide crucial information about the behavior and characteristics of the polynomial function.
Determining the zeros of a polynomial is a fundamental task in algebra, with applications spanning across various fields, including calculus, numerical analysis, and engineering. Understanding the zeros of a polynomial allows us to analyze its behavior, solve related equations, and model real-world phenomena. For instance, in physics, the zeros of a polynomial might represent the equilibrium points of a system, while in engineering, they could indicate the resonant frequencies of a circuit.
Evaluating p(x) at x = -1/√3
Now, let's turn our attention to the first value we need to verify: x = -1/√3. To determine whether this value is a zero of the polynomial p(x) = 3x² - 1, we must substitute x = -1/√3 into the polynomial expression and evaluate the result. This process involves replacing every instance of x in the polynomial with the value -1/√3 and then simplifying the expression.
Substituting x = -1/√3 into p(x), we get:
p(-1/√3) = 3(-1/√3)² - 1
To simplify this expression, we first need to square the term -1/√3. Squaring a negative number results in a positive number, and squaring a fraction involves squaring both the numerator and the denominator. Therefore:
(-1/√3)² = (-1)² / (√3)² = 1/3
Now, we can substitute this result back into our expression:
p(-1/√3) = 3(1/3) - 1
Next, we perform the multiplication:
p(-1/√3) = 1 - 1
Finally, we subtract:
p(-1/√3) = 0
As we can see, when we substitute x = -1/√3 into the polynomial p(x) = 3x² - 1, the result is 0. This confirms that x = -1/√3 is indeed a zero of the polynomial.
Evaluating p(x) at x = 2/√3
Next, we move on to the second value we need to verify: x = 2/√3. Following the same procedure as before, we substitute x = 2/√3 into the polynomial p(x) = 3x² - 1 and evaluate the result.
Substituting x = 2/√3 into p(x), we get:
p(2/√3) = 3(2/√3)² - 1
To simplify this expression, we first need to square the term 2/√3:
(2/√3)² = 2² / (√3)² = 4/3
Now, we substitute this result back into our expression:
p(2/√3) = 3(4/3) - 1
Next, we perform the multiplication:
p(2/√3) = 4 - 1
Finally, we subtract:
p(2/√3) = 3
In this case, when we substitute x = 2/√3 into the polynomial p(x) = 3x² - 1, the result is 3, which is not equal to 0. This indicates that x = 2/√3 is not a zero of the polynomial.
Conclusion: Zeros of p(x) = 3x² - 1
Through our detailed evaluation, we have successfully verified that x = -1/√3 is a zero of the polynomial p(x) = 3x² - 1, while x = 2/√3 is not. This process highlights the importance of direct substitution and evaluation in determining the zeros of a polynomial. By substituting the given values into the polynomial expression and simplifying, we can definitively confirm whether they satisfy the condition p(x) = 0. This understanding of polynomial zeros forms a crucial foundation for further exploration of polynomial behavior, root finding techniques, and their applications in various mathematical and scientific domains.
The polynomial p(x) = 3x² - 1 is a quadratic polynomial, and quadratic polynomials have at most two real roots. Our analysis has revealed one of these roots, x = -1/√3. To find the other root, we could employ various methods, such as factoring, using the quadratic formula, or applying numerical techniques. The zeros of a polynomial provide valuable information about its graph, its factors, and its relationship to other mathematical concepts. Further exploration of these concepts will deepen our understanding of polynomial functions and their significance in the broader mathematical landscape.
The study of polynomial functions is a cornerstone of mathematics, providing a framework for modeling a vast array of real-world phenomena. Among the key characteristics of polynomials, their zeros, also known as roots, hold particular significance. Zeros of a polynomial are the values of the variable that make the polynomial expression equal to zero. In this article, we will delve into the process of verifying whether specific values are zeros of the polynomial p(x) = 3x² - 1. We will meticulously examine the values x = -1/√3 and x = 2/√3, illustrating the fundamental techniques for determining polynomial roots. This exploration will not only enhance our understanding of polynomial zeros but also solidify our grasp of polynomial evaluation and root-finding methods.
Understanding Polynomials and Their Zeros
Before we embark on the verification process, let's establish a clear understanding of what polynomials are and why their zeros are so crucial. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The general form of a polynomial in one variable, x, is:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
where aₙ, aₙ₋₁, ..., a₁, a₀ are coefficients (constants) and n is a non-negative integer representing the degree of the polynomial. Polynomials are ubiquitous in mathematics and its applications, appearing in diverse fields such as physics, engineering, economics, and computer science. Their versatility stems from their ability to approximate complex functions and model intricate relationships.
The zeros of a polynomial, also referred to as roots, are the values of x for which p(x) = 0. These zeros are points where the graph of the polynomial intersects the x-axis. Finding the zeros of a polynomial is a fundamental problem in algebra with far-reaching implications. The zeros provide crucial insights into the behavior of the polynomial function, enabling us to solve equations, analyze stability, and design systems. For instance, in control theory, the zeros of a polynomial can determine the stability of a system, while in signal processing, they can represent the frequencies that are filtered out by a system.
The number of zeros a polynomial can have is directly related to its degree. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots, counting multiplicities. This means that a quadratic polynomial (degree 2) has at most two roots, a cubic polynomial (degree 3) has at most three roots, and so on. These roots can be real or complex numbers, and they may be distinct or repeated. The task of finding the roots of a polynomial can range from straightforward for simple polynomials to highly complex for higher-degree polynomials, often requiring numerical methods.
Verifying x = -1/√3 as a Zero of p(x) = 3x² - 1
Now, let's turn our attention to our specific polynomial, p(x) = 3x² - 1, and verify whether x = -1/√3 is a zero. To do this, we substitute x = -1/√3 into the polynomial expression and evaluate the result. If the result is zero, then x = -1/√3 is indeed a zero of the polynomial. This process is a direct application of the definition of a polynomial zero.
Substituting x = -1/√3 into p(x), we obtain:
p(-1/√3) = 3(-1/√3)² - 1
The next step is to simplify this expression. First, we square the term -1/√3. Remember that squaring a negative number results in a positive number, and squaring a fraction involves squaring both the numerator and the denominator:
(-1/√3)² = (-1)² / (√3)² = 1/3
Now, we substitute this result back into our expression:
p(-1/√3) = 3(1/3) - 1
Next, we perform the multiplication:
p(-1/√3) = 1 - 1
Finally, we subtract:
p(-1/√3) = 0
The result of our evaluation is 0. This conclusively demonstrates that x = -1/√3 is a zero of the polynomial p(x) = 3x² - 1. This means that if we were to plot the graph of this polynomial, it would intersect the x-axis at the point x = -1/√3. The fact that we found a zero indicates that * (x + 1/√3) * is a factor of the polynomial, which is a direct consequence of the Factor Theorem.
Determining if x = 2/√3 is a Zero of p(x) = 3x² - 1
Having verified that x = -1/√3 is a zero, let's now investigate whether x = 2/√3 is also a zero of the polynomial p(x) = 3x² - 1. We will employ the same method of direct substitution and evaluation as before. If substituting x = 2/√3 into the polynomial yields a result of 0, then it is a zero; otherwise, it is not.
Substituting x = 2/√3 into p(x), we get:
p(2/√3) = 3(2/√3)² - 1
To simplify this expression, we first square the term 2/√3:
(2/√3)² = 2² / (√3)² = 4/3
Now, we substitute this result back into our expression:
p(2/√3) = 3(4/3) - 1
Next, we perform the multiplication:
p(2/√3) = 4 - 1
Finally, we subtract:
p(2/√3) = 3
In this case, the result of substituting x = 2/√3 into p(x) is 3, which is not equal to 0. Therefore, we can definitively conclude that x = 2/√3 is not a zero of the polynomial p(x) = 3x² - 1. This means that the graph of the polynomial does not intersect the x-axis at the point x = 2/√3.
Conclusion: Identifying Zeros of Polynomial p(x) = 3x² - 1
Through our detailed analysis, we have successfully verified that x = -1/√3 is a zero of the polynomial p(x) = 3x² - 1, while x = 2/√3 is not. This exercise demonstrates the fundamental process of verifying polynomial zeros through direct substitution and evaluation. By substituting the given values into the polynomial expression and simplifying, we can determine whether they satisfy the condition p(x) = 0.
The polynomial p(x) = 3x² - 1 is a quadratic polynomial, and as we discussed earlier, quadratic polynomials have at most two real roots. We have identified one of these roots, x = -1/√3. To find the other root, we could use various methods, such as factoring, applying the quadratic formula, or employing numerical techniques. The roots of a polynomial provide valuable information about its graph, its factors, and its solutions to related equations. Further exploration of these concepts will deepen our understanding of polynomial functions and their applications.
In summary, the process of verifying polynomial zeros is a crucial skill in mathematics, enabling us to analyze the behavior of polynomial functions, solve equations, and model real-world phenomena. By mastering these techniques, we can unlock the power of polynomials and their applications across various scientific and engineering disciplines.
The study of polynomial equations is a fundamental aspect of algebra, with applications spanning diverse fields such as engineering, physics, and computer science. At the heart of solving polynomial equations lies the concept of roots, also known as zeros. A root of a polynomial equation is a value that, when substituted for the variable, makes the equation true. In this comprehensive exploration, we will meticulously examine the polynomial equation defined by p(x) = 3x² - 1 and determine whether the values x = -1/√3 and x = 2/√3 are indeed roots of this equation. Our investigation will provide a clear understanding of the process of root verification and its significance in the broader context of polynomial algebra.
The Significance of Roots in Polynomial Equations
Before we delve into the specifics of our example, it is crucial to understand the importance of roots in the context of polynomial equations. A polynomial equation is formed when a polynomial expression is set equal to zero. The roots of this equation are the values of the variable that satisfy the equation, meaning they make the left-hand side of the equation equal to the right-hand side (which is zero in this case).
Finding the roots of a polynomial equation is a central problem in mathematics. These roots represent the points where the graph of the polynomial intersects the x-axis, providing valuable insights into the behavior of the function. Moreover, roots are essential for solving various real-world problems that can be modeled using polynomial equations. For example, in physics, the roots of an equation might represent the equilibrium points of a system, while in engineering, they could indicate the resonant frequencies of a circuit. Understanding the roots of polynomial equations is, therefore, crucial for both theoretical and practical applications.
The number of roots a polynomial equation can have is determined by its degree, which is the highest power of the variable in the polynomial. According to the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n complex roots, counting multiplicities. This means that a quadratic equation (degree 2) has two roots, a cubic equation (degree 3) has three roots, and so on. These roots can be real or complex numbers, and they may be distinct or repeated. The challenge lies in finding these roots, as the methods for doing so vary depending on the degree and complexity of the polynomial equation.
Verifying x = -1/√3 as a Root of p(x) = 3x² - 1 = 0
Let's now focus on the polynomial equation p(x) = 3x² - 1 = 0 and verify whether x = -1/√3 is a root. To do this, we substitute x = -1/√3 into the equation and check if it satisfies the condition p(x) = 0. This process is a direct application of the definition of a root of a polynomial equation.
Substituting x = -1/√3 into p(x) = 3x² - 1, we get:
p(-1/√3) = 3(-1/√3)² - 1
Now, we simplify the expression step by step. First, we square the term -1/√3:
(-1/√3)² = (-1)² / (√3)² = 1/3
Next, we substitute this result back into our equation:
p(-1/√3) = 3(1/3) - 1
Then, we perform the multiplication:
p(-1/√3) = 1 - 1
Finally, we subtract:
p(-1/√3) = 0
The result of our evaluation is 0. This confirms that x = -1/√3 satisfies the equation p(x) = 3x² - 1 = 0, and therefore, it is indeed a root of the equation. This means that if we were to graph the polynomial p(x) = 3x² - 1, the graph would intersect the x-axis at the point x = -1/√3. Furthermore, the Factor Theorem tells us that (x + 1/√3) is a factor of the polynomial p(x).
Determining if x = 2/√3 is a Root of p(x) = 3x² - 1 = 0
Having verified that x = -1/√3 is a root, let's now investigate whether x = 2/√3 is also a root of the polynomial equation p(x) = 3x² - 1 = 0. We will follow the same procedure of substituting the value into the equation and evaluating the result.
Substituting x = 2/√3 into p(x) = 3x² - 1, we get:
p(2/√3) = 3(2/√3)² - 1
To simplify this expression, we first square the term 2/√3:
(2/√3)² = 2² / (√3)² = 4/3
Next, we substitute this result back into our equation:
p(2/√3) = 3(4/3) - 1
Then, we perform the multiplication:
p(2/√3) = 4 - 1
Finally, we subtract:
p(2/√3) = 3
In this case, the result of substituting x = 2/√3 into p(x) is 3, which is not equal to 0. This indicates that x = 2/√3 does not satisfy the equation p(x) = 3x² - 1 = 0, and therefore, it is not a root of the equation. This means that the graph of the polynomial p(x) = 3x² - 1 does not intersect the x-axis at the point x = 2/√3.
Conclusion: Identifying the Roots of Polynomial p(x) = 3x² - 1 = 0
Through our meticulous analysis, we have successfully verified that x = -1/√3 is a root of the polynomial equation p(x) = 3x² - 1 = 0, while x = 2/√3 is not. This process exemplifies the fundamental method of verifying roots of polynomial equations by direct substitution and evaluation. By substituting the given values into the equation and simplifying, we can definitively determine whether they satisfy the condition p(x) = 0.
The polynomial p(x) = 3x² - 1 is a quadratic polynomial, and as we know, quadratic equations have at most two roots. We have identified one of these roots, x = -1/√3. To find the other root, we could employ various techniques, such as factoring, applying the quadratic formula, or using numerical methods. The roots of a polynomial equation provide crucial information about the behavior of the polynomial function, its graph, and its solutions to related problems. Further investigation of these concepts will enhance our understanding of polynomial algebra and its diverse applications.
In conclusion, the verification of roots is a vital skill in mathematics, enabling us to solve polynomial equations, analyze the behavior of polynomial functions, and model real-world phenomena. By mastering these techniques, we can unlock the power of polynomials and their applications across various scientific and engineering disciplines.