Finding The Roots Of F(x) = X³ - 2x² - X + 2 A Step By Step Guide
Unlocking the secrets of polynomial equations is a cornerstone of algebra, and finding the roots of a cubic equation like f(x) = x³ - 2x² - x + 2 is a fundamental skill. This comprehensive guide will walk you through various methods to determine the roots of this equation, providing clarity and insights along the way. We will explore factoring techniques, the Rational Root Theorem, and synthetic division, ensuring you grasp the underlying concepts and can confidently solve similar problems.
Understanding the Problem: Finding the Roots
In mathematics, the roots of a function, also known as zeros, are the values of x that make the function equal to zero. In simpler terms, these are the points where the graph of the function intersects the x-axis. For the cubic equation f(x) = x³ - 2x² - x + 2, our mission is to find the values of x that satisfy the equation f(x) = 0. These roots provide critical information about the behavior of the polynomial and its graph.
Before diving into specific methods, it's essential to recognize the characteristics of a cubic equation. A cubic equation is a polynomial equation of degree three, meaning the highest power of the variable (in this case, x) is 3. According to the Fundamental Theorem of Algebra, a cubic equation has exactly three roots, which may be real or complex, and some roots may be repeated. This theorem serves as a guiding principle in our quest to find the roots of f(x).
To set the stage for our exploration, let's consider the options presented: A. 1, -2, -1, 2; B. -1, 1, 2; C. -1, 1, -2; D. -1, 1, -2, 0; E. -1, 1, 2, 0. We need to verify which of these sets, if any, correctly identifies the roots of f(x). Keep in mind that a cubic equation should have a maximum of three roots, so options with four values may be immediately suspect. With this foundational understanding, we're ready to delve into the methods for finding the roots of f(x).
Method 1: Factoring by Grouping – A Powerful Technique
Factoring by grouping is a powerful technique that can often simplify the process of finding roots, especially for polynomials with four terms. This method involves strategically grouping terms together and factoring out common factors. For our equation, f(x) = x³ - 2x² - x + 2, we can begin by grouping the first two terms and the last two terms:
(x³ - 2x²) + (-x + 2)
Now, we look for common factors within each group. In the first group, x³ - 2x², the common factor is x². Factoring this out, we get:
x²(x - 2)
In the second group, -x + 2, we can factor out a -1 to make the binomial consistent with the first group:
-1(x - 2)
Now, we rewrite the original expression with these factored groups:
x²(x - 2) - 1(x - 2)
Notice that (x - 2) is a common factor in both terms. This is the key to factoring by grouping. We factor out (x - 2) from the entire expression:
(x - 2)(x² - 1)
We've successfully factored the cubic into a product of a linear term (x - 2) and a quadratic term (x² - 1). Now, we can further factor the quadratic term. Recognize that x² - 1 is a difference of squares, which factors as (x - 1)(x + 1). Thus, our fully factored equation is:
(x - 2)(x - 1)(x + 1)
To find the roots, we set each factor equal to zero and solve for x:
- x - 2 = 0 => x = 2
- x - 1 = 0 => x = 1
- x + 1 = 0 => x = -1
Therefore, the roots of f(x) = x³ - 2x² - x + 2 are x = -1, 1, and 2. This matches option B in the provided choices, confirming that it is the correct answer. Factoring by grouping allowed us to efficiently break down the cubic equation into simpler factors, making it easy to identify the roots. This method highlights the power of algebraic manipulation in solving polynomial equations.
Method 2: The Rational Root Theorem – A Systematic Approach
The Rational Root Theorem provides a systematic way to identify potential rational roots of a polynomial equation. This theorem is particularly useful when factoring by grouping isn't immediately apparent or when dealing with polynomials that don't factor easily. The theorem states that if a polynomial equation with integer coefficients has rational roots, then these roots 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.
For our equation, f(x) = x³ - 2x² - x + 2, the constant term is 2, and the leading coefficient (the coefficient of x³) is 1. To apply the Rational Root Theorem, we first list the factors of the constant term (2) and the leading coefficient (1):
- Factors of the constant term (2): ±1, ±2
- Factors of the leading coefficient (1): ±1
Now, we form all possible rational roots by dividing each factor of the constant term by each factor of the leading coefficient:
Possible rational roots: ±1/±1, ±2/±1, which simplifies to ±1, ±2.
This gives us a set of potential roots: -2, -1, 1, and 2. The Rational Root Theorem narrows down the possibilities, allowing us to test these values in the original equation. We can substitute each potential root into f(x) to see if it results in zero:
- f(-2) = (-2)³ - 2(-2)² - (-2) + 2 = -8 - 8 + 2 + 2 = -12 ≠ 0
- f(-1) = (-1)³ - 2(-1)² - (-1) + 2 = -1 - 2 + 1 + 2 = 0
- f(1) = (1)³ - 2(1)² - (1) + 2 = 1 - 2 - 1 + 2 = 0
- f(2) = (2)³ - 2(2)² - (2) + 2 = 8 - 8 - 2 + 2 = 0
We find that f(-1) = 0, f(1) = 0, and f(2) = 0, which means -1, 1, and 2 are indeed the roots of the equation. This aligns with the roots we found using factoring by grouping and confirms the solution. The Rational Root Theorem provides a structured approach to identifying potential roots, making it an invaluable tool in solving polynomial equations.
Method 3: Synthetic Division – A Streamlined Approach for Root Verification
Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form (x - c). It's an efficient technique for verifying whether a potential root, identified through methods like the Rational Root Theorem, is indeed a root of the polynomial. Moreover, synthetic division provides the quotient polynomial, which can be helpful in further factoring or solving the equation.
To illustrate, let's use synthetic division to verify that x = -1 is a root of f(x) = x³ - 2x² - x + 2. We set up the synthetic division as follows:
-1 | 1 -2 -1 2
|
------------------
We bring down the leading coefficient (1):
-1 | 1 -2 -1 2
|
------------------
1
Multiply -1 by 1 and write the result under -2:
-1 | 1 -2 -1 2
| -1
------------------
1
Add -2 and -1:
-1 | 1 -2 -1 2
| -1
------------------
1 -3
Multiply -1 by -3 and write the result under -1:
-1 | 1 -2 -1 2
| -1 3
------------------
1 -3
Add -1 and 3:
-1 | 1 -2 -1 2
| -1 3
------------------
1 -3 2
Multiply -1 by 2 and write the result under 2:
-1 | 1 -2 -1 2
| -1 3 -2
------------------
1 -3 2
Add 2 and -2:
-1 | 1 -2 -1 2
| -1 3 -2
------------------
1 -3 2 0
The last number in the bottom row is the remainder. Since the remainder is 0, this confirms that x = -1 is a root of f(x). The other numbers in the bottom row (1, -3, 2) are the coefficients of the quotient polynomial, which is x² - 3x + 2. We can now factor this quadratic:
x² - 3x + 2 = (x - 1)(x - 2)
This gives us the roots x = 1 and x = 2, which, combined with x = -1, match the roots we found earlier. Synthetic division not only verifies roots but also aids in reducing the degree of the polynomial, making it easier to find all the roots. This technique provides an efficient and organized approach to polynomial division and root verification.
Conclusion: Mastering Root-Finding Techniques
In this comprehensive guide, we've explored three powerful methods for finding the roots of the cubic equation f(x) = x³ - 2x² - x + 2: factoring by grouping, the Rational Root Theorem, and synthetic division. Each method offers a unique approach and set of advantages, providing a versatile toolkit for solving polynomial equations.
Factoring by grouping allows us to break down polynomials into simpler factors by strategically grouping terms and identifying common factors. This method is particularly effective when the polynomial has four terms and exhibits a clear pattern for grouping.
The Rational Root Theorem provides a systematic way to identify potential rational roots by considering the factors of the constant term and the leading coefficient. This theorem narrows down the possibilities, making it easier to test potential roots.
Synthetic division is a streamlined method for verifying roots and reducing the degree of the polynomial. It offers an efficient way to divide a polynomial by a linear factor and obtain the quotient polynomial, which can be further factored to find additional roots.
By mastering these techniques, you can confidently tackle a wide range of polynomial equations and gain a deeper understanding of their behavior and properties. The roots of a polynomial are fundamental to understanding its graph, its solutions, and its applications in various mathematical and real-world contexts. Therefore, mastering these root-finding techniques is a valuable skill in algebra and beyond. Remember, the roots of f(x) = x³ - 2x² - x + 2 are -1, 1, and 2, and these methods provide the tools to discover them. The solution that is correct is B. -1, 1, 2.