Analyzing The Function H(x) = X^4 - 5x^3 + 11x^2 - 13x + 6 And Finding Its Zeros

In the realm of mathematics, polynomial functions hold a significant place, serving as fundamental building blocks for various mathematical models and applications. Among these functions, quartic polynomials, characterized by their highest degree term being x⁴, present a unique blend of complexity and elegance. In this comprehensive exploration, we delve into the intricacies of a specific quartic polynomial function, h(x) = x⁴ - 5x³ + 11x² - 13x + 6, and unravel its behavior, zeros, and properties. Our journey begins with the identification of a complex zero, 1 - √2i, a fascinating entry point into the world of complex numbers and their profound connection to polynomial roots. By meticulously dissecting the function and its zero, we aim to gain a deeper understanding of the underlying mathematical principles and their implications. This exploration will not only enhance our comprehension of polynomial functions but also showcase the beauty and interconnectedness of mathematical concepts.

The study of polynomial functions is not merely an academic exercise; it has far-reaching implications in various fields, including engineering, physics, and computer science. Polynomials are used to model a wide range of phenomena, from the trajectory of a projectile to the behavior of electrical circuits. Understanding the roots of a polynomial is crucial in many applications, as these roots often represent critical values or points of interest in the modeled system. For instance, in engineering, the roots of a polynomial may represent the resonant frequencies of a structure, while in physics, they may correspond to the energy levels of an atom. The ability to analyze and manipulate polynomial functions is therefore an essential skill for anyone working in these fields.

Our focus on the complex zero 1 - √2i highlights the importance of complex numbers in the study of polynomials. While the coefficients of our polynomial are real, its roots may include complex numbers, which are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1). Complex numbers may seem abstract, but they play a crucial role in many areas of mathematics and physics. In particular, the Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. This theorem guarantees that our quartic polynomial will have four roots, counting multiplicity, and that these roots may be real or complex. The presence of a complex zero in our polynomial function underscores the necessity of considering complex numbers when analyzing polynomial behavior.

The zeros of a polynomial function, also known as its roots, are the values of x for which the function evaluates to zero. In other words, they are the points where the graph of the function intersects the x-axis. Zeros play a crucial role in understanding the behavior of a polynomial function, as they provide valuable information about its factors, its symmetry, and its overall shape. For instance, if a polynomial function has a zero at x = a, then (x - a) must be a factor of the polynomial. This relationship between zeros and factors is a fundamental concept in polynomial algebra and forms the basis for many techniques used to solve polynomial equations.

In the case of our quartic polynomial h(x) = x⁴ - 5x³ + 11x² - 13x + 6, finding its zeros is a key objective. Knowing the zeros allows us to completely factor the polynomial, which in turn provides insights into its behavior. For example, the number of real zeros tells us how many times the graph of the function crosses the x-axis, and the location of the zeros gives us the intervals where the function is positive or negative. This information is invaluable in sketching the graph of the function and understanding its properties.

The given zero, 1 - √2i, is a complex number, which means it has both a real part (1) and an imaginary part (-√2). Complex zeros of polynomials with real coefficients always come in conjugate pairs. This means that if 1 - √2i is a zero of h(x), then its complex conjugate, 1 + √2i, must also be a zero. This property arises from the nature of complex numbers and their interaction with polynomial equations. The conjugate of a complex number a + bi is simply a - bi, where the sign of the imaginary part is changed. The fact that complex zeros occur in conjugate pairs simplifies the process of finding all the zeros of a polynomial, as we can deduce one zero from another.

The presence of complex zeros indicates that the graph of the polynomial function does not intersect the x-axis at the corresponding points. Complex zeros represent points where the function approaches zero in the complex plane, rather than on the real number line. This distinction highlights the importance of considering complex numbers when analyzing the behavior of polynomials, as they provide a complete picture of the function's roots.

The Conjugate Root Theorem is a powerful tool in polynomial algebra that states that if a polynomial with real coefficients has a complex zero a + bi, then its complex conjugate a - bi is also a zero. This theorem is a direct consequence of the properties of complex numbers and their behavior in polynomial equations. When a polynomial with real coefficients is evaluated at a complex number, the imaginary parts of the terms must cancel out in order for the result to be a real number. This cancellation can only occur if the complex conjugate is also a zero, ensuring that the imaginary parts cancel each other out.

In our case, since h(x) = x⁴ - 5x³ + 11x² - 13x + 6 has real coefficients and 1 - √2i is a zero, we can immediately conclude that 1 + √2i is also a zero. This significantly simplifies the task of finding all the zeros of the polynomial, as we now have two of the four zeros. Knowing two zeros allows us to factor out a quadratic factor from the polynomial, reducing the problem to finding the zeros of a simpler polynomial.

To find the quadratic factor corresponding to the zeros 1 - √2i and 1 + √2i, we can use the fact that if a and b are zeros of a polynomial, then (x - a) and (x - b) are factors of the polynomial. Therefore, the quadratic factor corresponding to these complex zeros is:

(x - (1 - √2i))(x - (1 + √2i))

Expanding this expression, we get:

x² - x(1 + √2i) - x(1 - √2i) + (1 - √2i)(1 + √2i)

Simplifying, we have:

x² - x - x√2i - x + x√2i + 1 + √2i - √2i - 2i²

Since i² = -1, the expression becomes:

x² - 2x + 1 + 2

Thus, the quadratic factor is:

x² - 2x + 3

This quadratic factor represents a significant step forward in our quest to fully factor the polynomial and determine all its zeros. By dividing the original quartic polynomial by this quadratic factor, we can obtain another quadratic polynomial, whose zeros will be the remaining zeros of the original polynomial.

Having identified the quadratic factor x² - 2x + 3 corresponding to the complex zeros, the next step is to divide the original polynomial h(x) = x⁴ - 5x³ + 11x² - 13x + 6 by this factor. Polynomial division is a fundamental technique in algebra that allows us to divide one polynomial by another, resulting in a quotient and a remainder. In this case, we expect the remainder to be zero, since x² - 2x + 3 is a factor of h(x). The quotient will be another quadratic polynomial, whose zeros will be the remaining zeros of h(x).

Performing polynomial long division, we have:

 x² - 3x + 2
 x² - 2x + 3 | x⁴ - 5x³ + 11x² - 13x + 6
 - (x⁴ - 2x³ + 3x²)
 ---------------------
 -3x³ + 8x² - 13x
 - (-3x³ + 6x² - 9x)
 ---------------------
 2x² - 4x + 6
 - (2x² - 4x + 6)
 ---------------------
 0

The result of the division is the quotient x² - 3x + 2 with a remainder of 0, as expected. This confirms that x² - 2x + 3 is indeed a factor of h(x), and we can now write h(x) as:

h(x) = (x² - 2x + 3)(x² - 3x + 2)

The next step is to factor the quadratic quotient x² - 3x + 2. This can be done by either factoring by inspection or using the quadratic formula. In this case, the quadratic factors easily by inspection:

x² - 3x + 2 = (x - 1)(x - 2)

Therefore, we have completely factored the polynomial h(x) as:

h(x) = (x² - 2x + 3)(x - 1)(x - 2)

This factorization provides us with all the zeros of h(x). The zeros are the values of x that make each factor equal to zero. From the factors (x - 1) and (x - 2), we obtain the real zeros x = 1 and x = 2. The zeros of the quadratic factor x² - 2x + 3 are the complex zeros we already identified, 1 - √2i and 1 + √2i.

With the polynomial h(x) = x⁴ - 5x³ + 11x² - 13x + 6 completely factored as (x² - 2x + 3)(x - 1)(x - 2), we can now confidently state all its zeros. The zeros are the values of x that make the polynomial equal to zero, which correspond to the roots of each factor.

From the linear factors (x - 1) and (x - 2), we obtain the real zeros:

• x = 1
• x = 2

From the quadratic factor x² - 2x + 3, we already know the complex zeros:

• x = 1 - √2i
• x = 1 + √2i

Therefore, the complete set of zeros for the polynomial h(x) is:

{1, 2, 1 - √2i, 1 + √2i}

This set includes two real zeros and two complex zeros, which is consistent with the Fundamental Theorem of Algebra, which states that a polynomial of degree n has n complex roots, counting multiplicity. In our case, h(x) is a quartic polynomial (degree 4), so it has four zeros.

The complete factorization of h(x) is:

h(x) = (x - 1)(x - 2)(x - (1 - √2i))(x - (1 + √2i))

This factorization expresses h(x) as a product of linear factors corresponding to each of its zeros. This form is particularly useful for analyzing the behavior of the polynomial, as it directly reveals the roots and their multiplicities. The linear factors (x - 1) and (x - 2) correspond to the real zeros 1 and 2, respectively, while the factors (x - (1 - √2i)) and (x - (1 + √2i)) correspond to the complex zeros 1 - √2i and 1 + √2i, respectively.

The complete factorization also allows us to easily reconstruct the original polynomial by expanding the product of the factors. This serves as a check on our work and confirms that we have correctly identified all the zeros and factors of h(x).

In this comprehensive exploration, we have successfully navigated the intricacies of the quartic polynomial function h(x) = x⁴ - 5x³ + 11x² - 13x + 6. We began with the identification of a complex zero, 1 - √2i, and utilized the Conjugate Root Theorem to deduce the presence of its conjugate, 1 + √2i, as another zero. This foundational step paved the way for factoring out a quadratic factor corresponding to these complex zeros, simplifying the polynomial and enabling further analysis.

Through polynomial division, we obtained the remaining quadratic factor, which, upon factorization, revealed the two real zeros, x = 1 and x = 2. This meticulous process culminated in the complete factorization of h(x), expressing it as a product of linear factors corresponding to each of its zeros. The complete set of zeros was determined to be {1, 2, 1 - √2i, 1 + √2i}, comprising two real zeros and two complex zeros, consistent with the Fundamental Theorem of Algebra.

This journey through the intricacies of a quartic polynomial function has not only reinforced our understanding of polynomial algebra but also highlighted the interconnectedness of mathematical concepts. The Conjugate Root Theorem, polynomial division, factorization techniques, and the Fundamental Theorem of Algebra all played crucial roles in unraveling the behavior of h(x). The presence of complex zeros underscored the importance of considering complex numbers in polynomial analysis, providing a complete picture of the function's roots.

The ability to analyze and manipulate polynomial functions is a valuable skill in various fields, including engineering, physics, and computer science. Polynomials are used to model a wide range of phenomena, and understanding their zeros is crucial for solving equations, optimizing systems, and making predictions. The techniques and concepts explored in this article provide a solid foundation for tackling more complex polynomial problems and applying them to real-world applications.

In conclusion, the exploration of h(x) = x⁴ - 5x³ + 11x² - 13x + 6 has been a rewarding endeavor, showcasing the beauty and power of polynomial algebra. By carefully dissecting the function, identifying its zeros, and factoring it completely, we have gained a deeper appreciation for the elegance and utility of mathematical principles. This understanding will undoubtedly serve as a valuable asset in future mathematical pursuits and applications.