Finding Roots Of Polynomial Function F(x) = X³ - 3x² + 2

Finding the roots of a polynomial function is a fundamental concept in algebra. In this comprehensive guide, we will delve into the process of identifying the roots of the polynomial function F(x) = x³ - 3x² + 2. We will explore various techniques and strategies to determine which of the given options, namely A. (2 - √12)/2, B. 1, C. (3 - √17)/4, and D. (3 + √17)/4, are indeed the roots of this function. Understanding the roots of a polynomial is crucial in many mathematical and scientific applications, as they represent the points where the function intersects the x-axis. Let's embark on this journey to unravel the intricacies of polynomial roots.

Understanding Polynomial Roots

Polynomial roots, also known as zeros or solutions, are the values of 'x' for which the polynomial function F(x) equals zero. In simpler terms, these are the points where the graph of the polynomial intersects the x-axis. Finding these roots is a critical skill in algebra, as it allows us to solve equations, factor polynomials, and understand the behavior of functions. For a cubic polynomial like F(x) = x³ - 3x² + 2, there can be up to three roots, which may be real or complex numbers. The quest to find these roots often involves a combination of algebraic techniques, including factoring, the rational root theorem, and numerical methods.

To effectively identify the roots, it's essential to grasp the concept of the Factor Theorem. This theorem states that if 'a' is a root of a polynomial F(x), then (x - a) is a factor of F(x). Conversely, if (x - a) is a factor of F(x), then 'a' is a root of F(x). This theorem provides a powerful tool for verifying potential roots and factoring polynomials. By systematically testing potential roots and applying the Factor Theorem, we can narrow down the possibilities and determine the actual roots of the polynomial. The roots play a pivotal role in understanding the graph of the polynomial, as they indicate where the graph crosses or touches the x-axis. A deep understanding of roots is invaluable for solving polynomial equations and analyzing their properties.

Techniques for Finding Roots

Several techniques can be employed to find the roots of a polynomial function. Factoring is one of the most straightforward methods, especially for simpler polynomials. If the polynomial can be factored into linear factors, each factor corresponds to a root. For example, if F(x) can be factored as (x - a)(x - b)(x - c), then the roots are a, b, and c. However, factoring is not always easy, especially for higher-degree polynomials or those with irrational or complex roots. In such cases, other techniques come into play.

The Rational Root Theorem is a valuable tool for identifying potential rational roots. This theorem states that if a polynomial has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By listing all possible rational roots using this theorem, we can systematically test them using synthetic division or direct substitution. If a potential root results in a remainder of zero, it is indeed a root of the polynomial. This method significantly narrows down the possibilities and helps us find rational roots efficiently. In addition to these algebraic methods, numerical methods such as the Newton-Raphson method can be used to approximate roots, especially when analytical solutions are difficult to obtain. These numerical techniques provide valuable approximations, allowing us to understand the behavior of the polynomial even when exact roots are elusive.

Applying Techniques to F(x) = x³ - 3x² + 2

Now, let's apply these techniques to the given polynomial function F(x) = x³ - 3x² + 2. First, we can attempt to factor the polynomial. However, simple factoring techniques may not readily reveal the factors. Therefore, we can turn to the Rational Root Theorem to identify potential rational roots. The constant term is 2, and the leading coefficient is 1. The factors of 2 are ±1 and ±2, and the factors of 1 are ±1. Thus, the possible rational roots are ±1 and ±2.

We can test these potential roots by substituting them into the polynomial. If F(x) = 0 for a particular value of x, then that value is a root. Let's start with x = 1: F(1) = (1)³ - 3(1)² + 2 = 1 - 3 + 2 = 0. This confirms that x = 1 is indeed a root of the polynomial. Now, we can use synthetic division or polynomial long division to divide F(x) by (x - 1) to find the remaining quadratic factor. Dividing x³ - 3x² + 2 by (x - 1) gives us the quadratic x² - 2x - 2. This quadratic can then be solved using the quadratic formula to find the remaining roots. By systematically applying these techniques, we can determine all the roots of the polynomial and understand its behavior more fully.

Testing the Given Options

To determine which of the given options are roots of F(x) = x³ - 3x² + 2, we need to substitute each option into the polynomial and check if the result is zero. This is a direct application of the definition of a root. Let's examine each option individually:

A. (2 - √12)/2: Substituting this value into F(x) involves calculating ((2 - √12)/2)³ - 3((2 - √12)/2)² + 2. This calculation can be simplified by first noting that √12 = 2√3, so the option becomes (1 - √3). Plugging this into F(x) and simplifying will reveal whether it results in zero. This process involves careful algebraic manipulation to avoid errors. The accuracy of the result is crucial in determining if this option is a root.

B. 1: We have already tested this value and found that F(1) = 0, so 1 is a root of the polynomial.

C. (3 - √17)/4: Substituting this value into F(x) requires calculating ((3 - √17)/4)³ - 3((3 - √17)/4)² + 2. This involves complex calculations with square roots and fractions. Simplifying the expression and checking if it equals zero will determine if this option is a root.

D. (3 + √17)/4: Similarly, substituting this value involves calculating ((3 + √17)/4)³ - 3((3 + √17)/4)² + 2. This calculation is similar to option C but with a positive square root term. Again, careful simplification and verification are necessary to determine if this option is a root.

By methodically substituting each option and verifying if the result is zero, we can identify the roots of the polynomial function. This process demonstrates the practical application of the definition of a root and the importance of accurate algebraic manipulation.

Detailed Analysis of the Options

Let's delve into a more detailed analysis of each option to determine which are the roots of the polynomial function F(x) = x³ - 3x² + 2. This involves careful calculations and algebraic manipulations.

Option A: (2 - √12)/2

As noted earlier, we can simplify this option to (1 - √3). Substituting this into F(x), we get:

F(1 - √3) = (1 - √3)³ - 3(1 - √3)² + 2

Expanding this expression:

= (1 - 3√3 + 9 - 3√3) - 3(1 - 2√3 + 3) + 2

= (10 - 6√3) - 3(4 - 2√3) + 2

= 10 - 6√3 - 12 + 6√3 + 2

= 0

Therefore, (2 - √12)/2 or (1 - √3) is a root of the polynomial.

Option B: 1

We have already verified that F(1) = 0, so 1 is a root.

Option C: (3 - √17)/4

Substituting this value into F(x) involves more complex calculations. Let's denote (3 - √17)/4 as 'r'. Then:

F(r) = r³ - 3r² + 2

r³ = ((3 - √17)/4)³ = (27 - 27√17 + 17*3 - 17√17) / 64 = (78 - 44√17)/64

3r² = 3((3 - √17)/4)² = 3(9 - 6√17 + 17) / 16 = 3(26 - 6√17) / 16 = (78 - 18√17) / 16

Now, we substitute these values into F(r):

F(r) = (78 - 44√17)/64 - (78 - 18√17)/16 + 2

To simplify, we find a common denominator of 64:

F(r) = (78 - 44√17 - 4*(78 - 18√17) + 128) / 64

= (78 - 44√17 - 312 + 72√17 + 128) / 64

= (-106 + 28√17) / 64

This expression is not equal to zero, so (3 - √17)/4 is not a root.

Option D: (3 + √17)/4

The process for this option is similar to option C. Let's denote (3 + √17)/4 as 's'. Then:

F(s) = s³ - 3s² + 2

s³ = ((3 + √17)/4)³ = (27 + 27√17 + 17*3 + 17√17) / 64 = (78 + 44√17) / 64

3s² = 3((3 + √17)/4)² = 3(9 + 6√17 + 17) / 16 = 3(26 + 6√17) / 16 = (78 + 18√17) / 16

Now, we substitute these values into F(s):

F(s) = (78 + 44√17) / 64 - (78 + 18√17) / 16 + 2

To simplify, we find a common denominator of 64:

F(s) = (78 + 44√17 - 4*(78 + 18√17) + 128) / 64

= (78 + 44√17 - 312 - 72√17 + 128) / 64

= (-106 - 28√17) / 64

This expression is not equal to zero, so (3 + √17)/4 is not a root.

Conclusion

In conclusion, after a detailed analysis and calculation, we have determined that the roots of the polynomial function F(x) = x³ - 3x² + 2 among the given options are:

• (A) (2 - √12) / 2
• (B) 1

Options (C) and (D) were found not to be roots of the polynomial. This process highlights the importance of using various techniques such as the Rational Root Theorem, synthetic division, and direct substitution to find the roots of polynomial functions. Understanding these methods is crucial for solving algebraic problems and gaining a deeper insight into the behavior of polynomials. The ability to accurately identify roots is a fundamental skill in mathematics with applications in various fields, including engineering, physics, and computer science.