Finding Roots Of Polynomial Function F(x) = X³ - 5x² - 13x - 7

In this comprehensive article, we delve into the intricate process of identifying the roots of a polynomial function. Specifically, we will focus on the function F(x) = x³ - 5x² - 13x - 7. Our objective is to meticulously determine which of the provided options – A. 3 - √2, B. 7, C. 3 + √2, D. 1 - √3, E. -1, and F. 1 + √3 – are indeed the roots of this cubic polynomial. Understanding the roots of a polynomial is a fundamental concept in algebra, with far-reaching implications in various fields such as engineering, physics, and computer science. A root of a polynomial, also known as a zero, is a value of x that makes the polynomial equal to zero. Finding these roots is essential for solving equations, graphing functions, and analyzing the behavior of systems modeled by polynomials. In this exploration, we will employ a combination of algebraic techniques and the Rational Root Theorem to systematically evaluate each potential root and determine its validity.

Understanding Polynomial Roots

To begin, it is crucial to grasp the essence of polynomial roots. A root, or zero, of a polynomial function F(x) is a value x = a such that F(a) = 0. Graphically, these roots correspond to the points where the polynomial's graph intersects the x-axis. For a cubic polynomial like the one we are examining, there can be up to three roots, considering both real and complex solutions. The Fundamental Theorem of Algebra guarantees that a polynomial of degree n will have exactly n complex roots, counted with multiplicity. This means that some roots may be repeated. Identifying these roots involves a blend of algebraic manipulation, strategic substitution, and, when necessary, the application of theorems like the Rational Root Theorem and the Conjugate Root Theorem.

The Rational Root Theorem is particularly useful when dealing with polynomials with integer coefficients. It states that if a polynomial has a rational root p/q (where p and q are integers with no common factors), then p must be a factor of the constant term, and q must be a factor of the leading coefficient. In our case, the polynomial F(x) = x³ - 5x² - 13x - 7 has a leading coefficient of 1 and a constant term of -7. Therefore, any rational roots must be factors of -7, which are ±1 and ±7. This theorem narrows down the possibilities we need to check, making the process more efficient. Additionally, the Conjugate Root Theorem is relevant when dealing with polynomials with real coefficients. It asserts that if a polynomial has a complex root of the form a + bi, then its complex conjugate a - bi is also a root. This theorem extends to irrational roots of the form a + √b, where a - √b would also be a root. Understanding these theorems provides a robust framework for finding the roots of polynomial functions, enabling us to approach the problem systematically and with confidence.

Applying the Rational Root Theorem

To effectively identify the roots of the polynomial F(x) = x³ - 5x² - 13x - 7, we will first apply the Rational Root Theorem. This theorem significantly narrows down the potential rational roots, making the process of verification much more manageable. As mentioned earlier, the Rational Root Theorem states that any rational root p/q of a polynomial with integer coefficients must have p as a factor of the constant term and q as a factor of the leading coefficient. In our polynomial, the constant term is -7, and the leading coefficient is 1. The factors of -7 are ±1 and ±7, while the factors of 1 are ±1. Therefore, the possible rational roots are ±1 and ±7. We will now systematically test these values to see if they satisfy the equation F(x) = 0.

First, let's test x = 1: F(1) = (1)³ - 5(1)² - 13(1) - 7 = 1 - 5 - 13 - 7 = -24. Since F(1) ≠ 0, 1 is not a root. Next, we test x = -1: F(-1) = (-1)³ - 5(-1)² - 13(-1) - 7 = -1 - 5 + 13 - 7 = 0. Thus, x = -1 is a root of the polynomial. This is a significant finding, as it allows us to factor the polynomial and potentially find other roots more easily. Now, we test x = 7: F(7) = (7)³ - 5(7)² - 13(7) - 7 = 343 - 245 - 91 - 7 = 0. Therefore, x = 7 is also a root of the polynomial. With two roots identified, we can proceed to further factor the polynomial and determine if there are any additional roots. This systematic approach, guided by the Rational Root Theorem, has proven to be an efficient way to narrow down the possibilities and find the rational roots of the polynomial.

Testing the Potential Roots

Having applied the Rational Root Theorem and identified -1 and 7 as rational roots, we now turn our attention to the remaining options: A. 3 - √2, C. 3 + √2, D. 1 - √3, and F. 1 + √3. These roots are irrational, and the Conjugate Root Theorem will be instrumental in their evaluation. The Conjugate Root Theorem states that if a polynomial with real coefficients has an irrational root of the form a + √b, then its conjugate a - √b is also a root. This means that if 3 + √2 is a root, then 3 - √2 must also be a root, and similarly, if 1 + √3 is a root, then 1 - √3 must also be a root.

To test these potential irrational roots, we can directly substitute them into the polynomial F(x) = x³ - 5x² - 13x - 7. Let's start with 3 + √2: F(3 + √2) = (3 + √2)³ - 5(3 + √2)² - 13(3 + √2) - 7. Expanding this expression is complex, but if it equals zero, then 3 + √2 is a root. Due to the complexity of manual calculation, it's often more efficient to use synthetic division or polynomial division once a root is known. Since we've identified -1 as a root, we can divide F(x) by (x + 1) to obtain a quadratic expression. This quadratic can then be solved using the quadratic formula or factoring to find the remaining roots. Let's perform the polynomial division: Dividing x³ - 5x² - 13x - 7 by (x + 1) yields x² - 6x - 7. Now, we can solve the quadratic equation x² - 6x - 7 = 0. This can be factored as (x - 7)(x + 1) = 0, which gives us the roots x = 7 and x = -1, as we already found. This indicates that the polynomial can be written as F(x) = (x + 1)(x - 7)(x + 1), showing that -1 is a repeated root. Now, let's consider the remaining options. If 3 + √2 or 1 + √3 were roots, they would appear as solutions to the quadratic x² - 6x - 7 = 0. However, the roots of this quadratic are -1 and 7, which are rational. Thus, 3 + √2 and 1 + √3 are not roots of the original polynomial. This systematic evaluation, combining the Conjugate Root Theorem and polynomial division, allows us to accurately determine which irrational numbers are roots of the given polynomial.

Final Roots and Conclusion

After a thorough examination using the Rational Root Theorem, Conjugate Root Theorem, and polynomial division, we have definitively identified the roots of the polynomial function F(x) = x³ - 5x² - 13x - 7. Our analysis revealed that the rational roots are x = -1 and x = 7. The root x = -1 has a multiplicity of 2, meaning it appears twice as a root, while x = 7 is a single root. The other options provided, namely 3 - √2, 3 + √2, 1 - √3, and 1 + √3, were tested and found not to be roots of the polynomial.

In conclusion, the roots of the polynomial function F(x) = x³ - 5x² - 13x - 7 are -1 and 7. This determination was achieved through a systematic application of key algebraic principles and theorems. The Rational Root Theorem allowed us to narrow down the possible rational roots, and the Conjugate Root Theorem guided our understanding of potential irrational roots. Polynomial division, combined with factoring techniques, enabled us to break down the cubic polynomial into simpler factors, revealing the roots and their multiplicities. Understanding the roots of polynomial functions is crucial in various mathematical and scientific contexts, and the methods employed in this exploration provide a robust framework for tackling similar problems. This comprehensive approach underscores the importance of combining theoretical knowledge with practical techniques to effectively solve complex algebraic challenges.