Hey guys! Today, we're diving into the fascinating world of polynomial functions, specifically focusing on how to identify rational roots. We'll be using the function f(x) = 2x³ - 19x² + 57x - 54 as our case study. The graph of this cubic function is provided, and our mission is to determine just how many of its roots are rational numbers. Buckle up, because we're about to embark on a mathematical journey!
Understanding the Problem: Rational Roots and the Graph
So, what exactly are rational roots, and how do they relate to the graph of a function? A rational root of a polynomial function is a root that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Essentially, these are the x-values where the graph of the function intersects the x-axis, and these values can be written as simple fractions or whole numbers. When we're looking at the graph, these intersections are visually apparent, but it's our job to determine if the corresponding x-values are indeed rational.
Now, let’s talk about our specific function, f(x) = 2x³ - 19x² + 57x - 54. This is a cubic function, meaning it has a degree of 3. A fundamental theorem in algebra tells us that a polynomial function of degree n will have exactly n complex roots, counting multiplicities. This means our cubic function will have three roots in total. These roots can be real or complex, and the real roots are where the graph crosses or touches the x-axis. Our task is to pinpoint how many of these real roots are rational numbers.
Looking at the provided graph, we can visually identify the points where the function intersects the x-axis. These points represent the real roots of the function. The key is to examine these points closely and determine whether their x-values can be expressed as a fraction of two integers. Sometimes, the graph will clearly show an intersection at a whole number, which is obviously rational. Other times, the intersection might appear to be at a fractional value. This is where we need to use algebraic techniques, specifically the Rational Root Theorem, to confirm whether these apparent fractional roots are indeed rational.
The Rational Root Theorem is our powerful tool in this investigation. It provides a systematic way to list all possible rational roots of a polynomial function. The theorem states that if a polynomial has integer coefficients, then every rational root of the polynomial has the form p/q, where p is a factor of the constant term (the term without any x) and q is a factor of the leading coefficient (the coefficient of the highest power of x). By applying this theorem, we can create a list of potential rational roots and then test each one to see if it's actually a root of the function. This testing can be done by substituting the potential root into the function and checking if the result is zero, or by using synthetic division to see if the remainder is zero.
By combining the visual information from the graph with the algebraic rigor of the Rational Root Theorem, we can confidently determine the number of rational roots of our cubic function. It's like being a mathematical detective, using clues from the graph and the theorem to solve the mystery of the roots!
Applying the Rational Root Theorem
Alright, let’s get down to business and apply the Rational Root Theorem to our function, f(x) = 2x³ - 19x² + 57x - 54. Remember, the theorem helps us identify potential rational roots by considering the factors of the constant term and the leading coefficient.
First, we need to identify the constant term and the leading coefficient. In our function, the constant term is -54, and the leading coefficient is 2. Now, we’ll list out the factors of each:
- Factors of -54 (p): ±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54
- Factors of 2 (q): ±1, ±2
The Rational Root Theorem tells us that any rational root of f(x) must be of the form p/q, where p is a factor of -54 and q is a factor of 2. So, we’ll form all possible fractions using the factors we listed:
Possible Rational Roots (p/q):
- ±1/1, ±2/1, ±3/1, ±6/1, ±9/1, ±18/1, ±27/1, ±54/1
- ±1/2, ±2/2, ±3/2, ±6/2, ±9/2, ±18/2, ±27/2, ±54/2
Simplifying these fractions and removing duplicates, we get our list of potential rational roots:
±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54, ±1/2, ±3/2, ±9/2, ±27/2
That’s quite a list! Now, we need to test each of these potential roots to see if they actually make the function equal to zero. We can do this by substituting each value into f(x) and evaluating, or we can use synthetic division, which is often a faster method.
Let's start by testing some of the simpler integer values. For instance, we can try 1, -1, 2, and so on. If we substitute x = 2 into f(x), we get:
f(2) = 2(2)³ - 19(2)² + 57(2) - 54 = 16 - 76 + 114 - 54 = 0
Bingo! x = 2 is a rational root. This means (x - 2) is a factor of our polynomial. Now, we can use synthetic division to divide f(x) by (x - 2) and find the remaining quadratic factor.
Synthetic division is a neat way to divide polynomials. We set up the division using the coefficients of our polynomial (2, -19, 57, -54) and the root we’re testing (2). The process involves bringing down the first coefficient, multiplying it by the root, adding it to the next coefficient, and repeating until we reach the end. The last number we get is the remainder, and the other numbers are the coefficients of the quotient polynomial. If the remainder is zero, it confirms that the value we tested is indeed a root.
After performing synthetic division with x = 2, we’ll obtain a quadratic factor. This quadratic can then be solved using the quadratic formula or by factoring (if possible) to find the remaining roots of the cubic function. These roots will tell us the final answer to how many rational roots the function has.
Finding the Roots and the Answer
Okay, let's continue our quest to find all the roots of f(x) = 2x³ - 19x² + 57x - 54. As we discovered earlier, x = 2 is a rational root. Now, we'll use synthetic division to divide f(x) by (x - 2) and find the resulting quadratic factor. This will help us determine the remaining roots.
Here’s how the synthetic division looks:
2 | 2 -19 57 -54
| 4 -30 54
------------------
2 -15 27 0
The numbers on the bottom row (2, -15, 27) are the coefficients of the quotient polynomial, which is a quadratic: 2x² - 15x + 27. The remainder is 0, which confirms that x = 2 is indeed a root and (x - 2) is a factor of f(x).
Now, we need to find the roots of this quadratic equation, 2x² - 15x + 27 = 0. We can use the quadratic formula to do this. Remember the quadratic formula? It’s:
x = (-b ± √(b² - 4ac)) / (2a)
In our quadratic equation, a = 2, b = -15, and c = 27. Plugging these values into the quadratic formula, we get:
x = (15 ± √((-15)² - 4 * 2 * 27)) / (2 * 2)
x = (15 ± √(225 - 216)) / 4
x = (15 ± √9) / 4
x = (15 ± 3) / 4
This gives us two possible solutions:
- x = (15 + 3) / 4 = 18 / 4 = 9/2
- x = (15 - 3) / 4 = 12 / 4 = 3
So, the roots of the quadratic equation are x = 9/2 and x = 3. Both of these are rational numbers since they can be expressed as fractions.
Now, let's gather all the roots we’ve found:
- x = 2 (from our initial testing and synthetic division)
- x = 9/2 (from solving the quadratic equation)
- x = 3 (from solving the quadratic equation)
We have found three roots in total, and all three of them (2, 9/2, and 3) are rational numbers. Therefore, the cubic function f(x) = 2x³ - 19x² + 57x - 54 has three rational roots.
Looking back at the graph, we can visually confirm that the function intersects the x-axis at three points, and our algebraic analysis has shown that these points correspond to the rational roots we've calculated. It's always a good idea to cross-reference our algebraic results with the graph to ensure everything aligns.
So, to answer the original question: How many roots of f(x) = 2x³ - 19x² + 57x - 54 are rational numbers? The answer is 3. We successfully navigated through the problem by applying the Rational Root Theorem, performing synthetic division, and using the quadratic formula. Great job, guys! You've just tackled a challenging problem involving polynomial functions and rational roots.
Conclusion: The Power of Combining Visual and Algebraic Techniques
In this exploration, we've seen how powerful it is to combine visual information from a graph with algebraic techniques like the Rational Root Theorem and synthetic division. By analyzing the graph of the cubic function f(x) = 2x³ - 19x² + 57x - 54, we got a visual sense of the roots, which guided our algebraic work. The Rational Root Theorem then gave us a systematic way to identify potential rational roots, and synthetic division helped us to efficiently test these possibilities and factor the polynomial.
We successfully determined that the function has three rational roots: 2, 3, and 9/2. This journey demonstrates the importance of having a toolkit of mathematical methods at your disposal. When faced with a problem, you can draw upon different techniques and perspectives to arrive at the solution. The combination of graphical analysis and algebraic manipulation is a common and effective strategy in mathematics.
Understanding the relationship between the graph of a function and its roots is a fundamental concept in algebra. The roots, or zeros, of a function are the x-values where the function's graph intersects the x-axis. These roots are crucial in many applications, such as solving equations, modeling real-world phenomena, and understanding the behavior of functions. The Rational Root Theorem is a valuable tool for finding rational roots, but it's just one piece of the puzzle. There are other methods for finding roots, such as numerical techniques, and sometimes the roots might be irrational or complex numbers.
This problem also highlights the importance of accuracy and attention to detail in mathematical work. A small mistake in the synthetic division or the quadratic formula can lead to incorrect roots. So, it’s essential to double-check your calculations and make sure you're applying the techniques correctly. Practice is key to mastering these skills!
I hope you’ve enjoyed this deep dive into finding rational roots. Remember, guys, mathematics is like a puzzle, and the more tools and techniques you learn, the better you'll become at solving those puzzles. Keep exploring, keep questioning, and keep learning! You've got the power to tackle any mathematical challenge that comes your way. And remember, understanding these concepts not only helps in academics but also sharpens your analytical and problem-solving skills, which are valuable in all aspects of life. So, keep up the great work, and I'll see you in the next mathematical adventure!