Finding Actual Roots Using The Rational Root Theorem For F(x)=6x^4+5x^3-33x^2-12x+20
Potential roots of a polynomial are crucial in finding the actual roots, and the Rational Root Theorem provides a systematic way to identify these potential roots. In this article, we delve into the application of the Rational Root Theorem to the polynomial function f(x) = 6x⁴ + 5x³ - 33x² - 12x + 20. We are given a set of potential roots: -5/2, -2, 1, and 10/3. Our task is to determine which of these is an actual root of f(x). To accomplish this, we will meticulously evaluate f(x) at each potential root, and any value that yields f(x) = 0 confirms the root. Understanding the intricacies of polynomial roots is fundamental in algebra and calculus, with applications spanning diverse fields such as engineering, physics, and computer science. Thus, the ability to identify and verify potential roots is an indispensable skill for anyone working with polynomial functions.
Verifying Potential Roots
In order to verify which of the potential roots are actual roots, we will substitute each potential root into the polynomial function f(x) = 6x⁴ + 5x³ - 33x² - 12x + 20 and check if the result is zero. This process involves careful arithmetic and a methodical approach. First, let's consider the potential root -5/2. Substituting this value into f(x) gives us f(-5/2) = 6(-5/2)⁴ + 5(-5/2)³ - 33(-5/2)² - 12(-5/2) + 20. We need to compute each term meticulously. The calculation involves raising fractions to various powers, multiplying by coefficients, and combining like terms. If the result of this calculation is zero, then -5/2 is an actual root of the polynomial. If it's not zero, we move on to the next potential root. Next, we consider the potential root -2. We calculate f(-2) = 6(-2)⁴ + 5(-2)³ - 33(-2)² - 12(-2) + 20. Again, we carefully evaluate each term. If the result is zero, then -2 is an actual root. If not, we proceed to the next potential root. We repeat this process for the potential roots 1 and 10/3. For 1, we compute f(1) = 6(1)⁴ + 5(1)³ - 33(1)² - 12(1) + 20. For 10/3, we compute f(10/3) = 6(10/3)⁴ + 5(10/3)³ - 33(10/3)² - 12(10/3) + 20. Each calculation is critical to correctly identify the actual roots.
Detailed Calculations
Let's perform the detailed calculations to determine which potential root is an actual root of f(x). First, we evaluate f(-5/2):
f(-5/2) = 6(-5/2)⁴ + 5(-5/2)³ - 33(-5/2)² - 12(-5/2) + 20
f(-5/2) = 6(625/16) + 5(-125/8) - 33(25/4) + 30 + 20
f(-5/2) = 3750/16 - 625/8 - 825/4 + 50
To simplify, we convert all fractions to have a common denominator of 16:
f(-5/2) = 3750/16 - 1250/16 - 3300/16 + 800/16
f(-5/2) = (3750 - 1250 - 3300 + 800)/16
f(-5/2) = (4550 - 4550)/16
f(-5/2) = 0/16
f(-5/2) = 0
Since f(-5/2) = 0, we conclude that -5/2 is an actual root of f(x). Now, let's evaluate f(-2):
f(-2) = 6(-2)⁴ + 5(-2)³ - 33(-2)² - 12(-2) + 20
f(-2) = 6(16) + 5(-8) - 33(4) + 24 + 20
f(-2) = 96 - 40 - 132 + 24 + 20
f(-2) = 140 - 172
f(-2) = -32
Since f(-2) ≠ 0, -2 is not an actual root. Next, we evaluate f(1):
f(1) = 6(1)⁴ + 5(1)³ - 33(1)² - 12(1) + 20
f(1) = 6 + 5 - 33 - 12 + 20
f(1) = 31 - 45
f(1) = -14
Since f(1) ≠ 0, 1 is not an actual root. Finally, we evaluate f(10/3):
f(10/3) = 6(10/3)⁴ + 5(10/3)³ - 33(10/3)² - 12(10/3) + 20
f(10/3) = 6(10000/81) + 5(1000/27) - 33(100/9) - 12(10/3) + 20
f(10/3) = 60000/81 + 5000/27 - 3300/9 - 40 + 20
To simplify, we convert all fractions to have a common denominator of 81:
f(10/3) = 60000/81 + 15000/81 - 29700/81 - 3240/81 + 1620/81
f(10/3) = (60000 + 15000 - 29700 - 3240 + 1620)/81
f(10/3) = (76620 - 32940)/81
f(10/3) = 43680/81
Since f(10/3) ≠ 0, 10/3 is not an actual root. Therefore, after evaluating all potential roots, we find that only -5/2 is an actual root of f(x).
The Rational Root Theorem
The Rational Root Theorem is a cornerstone in algebra for identifying potential rational roots of a polynomial. This theorem is invaluable when seeking the roots of a polynomial equation, particularly when factoring or other root-finding methods are not immediately apparent. The theorem provides a methodical approach to narrow down the possible candidates for rational roots, making the search process more efficient. At its core, the Rational Root Theorem states that if a polynomial has a rational root p/q, where p and q are coprime integers, then p must be a factor of the constant term of the polynomial, and q must be a factor of the leading coefficient. This principle is powerful because it transforms the problem of finding roots into a more manageable task of checking a finite set of possible values. Applying the Rational Root Theorem begins with identifying the constant term and the leading coefficient of the polynomial. The factors of each of these terms are then listed. By forming all possible fractions p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, we generate a list of potential rational roots. Each potential root can then be tested by substituting it into the polynomial. If the result is zero, then that potential root is an actual root of the polynomial. The Rational Root Theorem does not guarantee that a polynomial has rational roots; it only provides a list of candidates to test. If none of the candidates are actual roots, it means the polynomial either has irrational roots, complex roots, or no real roots. Despite this limitation, the theorem is a powerful tool for simplifying the root-finding process, particularly in the context of algebraic problem-solving and polynomial analysis.
Practical Application
Applying the Rational Root Theorem in practice involves a systematic approach. Consider a polynomial equation, such as the one we are examining, f(x) = 6x⁴ + 5x³ - 33x² - 12x + 20. To apply the theorem, we first identify the constant term and the leading coefficient. In this case, the constant term is 20 and the leading coefficient is 6. Next, we list all the factors of these two numbers. The factors of 20 are ±1, ±2, ±4, ±5, ±10, and ±20. The factors of 6 are ±1, ±2, ±3, and ±6. According to the Rational Root Theorem, any rational root of f(x) must be of the form p/q, where p is a factor of 20 and q is a factor of 6. We then form all possible fractions using these factors. This results in a list of potential rational roots. The list might seem extensive, but it is a finite set of values, which significantly narrows down the possibilities. Some of the potential roots include ±1, ±2, ±4, ±5, ±10, ±20, ±1/2, ±5/2, ±1/3, ±2/3, ±4/3, ±5/3, ±10/3, ±20/3, ±1/6, and ±5/6. Once we have the list of potential roots, we can test each one by substituting it into the polynomial f(x). If f(p/q) = 0, then p/q is an actual root of the polynomial. This process often involves some tedious calculations, but it is straightforward. In the example we are working with, the potential roots provided (-5/2, -2, 1, 10/3) are derived from this process. By systematically testing these values, as we demonstrated earlier, we can identify the actual rational roots of the polynomial. The practical application of the Rational Root Theorem is not only useful in academic settings but also in real-world problems where polynomial equations arise, such as in engineering, physics, and economics.
Conclusion
In conclusion, by systematically applying the Rational Root Theorem and verifying each potential root through direct substitution into the polynomial f(x) = 6x⁴ + 5x³ - 33x² - 12x + 20, we have determined that -5/2 is the actual root among the given options. The Rational Root Theorem provides a structured approach to narrow down the possibilities, and the careful evaluation of each potential root is crucial for accurate results. This process underscores the importance of methodical problem-solving in mathematics and highlights the power of algebraic theorems in simplifying complex calculations. The ability to identify and verify roots of polynomials is a fundamental skill with broad applications in various fields, making a thorough understanding of these concepts essential for students and professionals alike.
