Rational Roots Of F(x) = 2x³ - 19x² + 57x - 54

Hey guys! Today, we're diving deep into the fascinating world of polynomial functions, specifically focusing on the cubic function f(x) = 2x³ - 19x² + 57x - 54. We've got its graph right here, and our mission is to figure out just how many of its roots are rational numbers. So, buckle up and let's get started!

Understanding the Problem: Rational Roots and the Graph

Before we jump into solving, let's make sure we're all on the same page. When we talk about the roots of a function, we're essentially asking: For what values of x does f(x) equal zero? Graphically, these roots are the points where the graph of the function intersects the x-axis. These intersection points, where the function's value is zero, hold significant importance in understanding the function's behavior and properties. Now, the term "rational" adds another layer to our quest. A rational number is simply a number that can be expressed as a fraction p/q, where p and q are integers (and q isn't zero, of course). Think of numbers like 1/2, -3/4, 5 (which is 5/1), and even 0 (which is 0/1). So, our main objective now is to identify how many of those x-intercepts, the roots, fall into this category of rational numbers. By carefully examining the given graph of f(x) = 2x³ - 19x² + 57x - 54, we're looking for points where the curve crosses the x-axis at locations that correspond to rational values. This involves a blend of visual analysis and algebraic understanding, where the graph provides a visual guide and our knowledge of rational numbers helps in pinpointing the exact roots. This graphical method is a powerful tool in mathematics, offering an intuitive way to grasp the nature and behavior of functions and their roots. Analyzing the graph, we're essentially embarking on a mathematical scavenger hunt, where the prize is the knowledge of the function's rational roots. This blend of visual and analytical techniques makes the process not just a mathematical exercise, but also an engaging intellectual pursuit.

The Rational Root Theorem: Our Detective Tool

Now, staring at the graph can give us some clues, but to be absolutely sure, we need a more systematic approach. That's where the Rational Root Theorem comes to the rescue! This theorem is like a detective's magnifying glass for polynomial functions. It gives us a list of potential rational roots, narrowing down our search significantly. So, how does this magical theorem work? The Rational Root Theorem states that if a polynomial has integer coefficients (which our f(x) definitely does), then any rational root p/q (in simplest form) must have p as a factor of the constant term and q as a factor of the leading coefficient. Let's break this down for our specific function, f(x) = 2x³ - 19x² + 57x - 54. The constant term is -54, and the leading coefficient (the coefficient of the x³ term) is 2. Now, we need to list all the factors of -54 and 2. Factors of -54: ±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54. Factors of 2: ±1, ±2. According to the Rational Root Theorem, any rational root of our function must be in the form of (factor of -54) / (factor of 2). This means our list of potential rational roots is: ±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54, ±1/2, ±3/2, ±9/2, ±27/2. Whoa! That's a hefty list, but it's still a finite list. This theorem has transformed our problem from an infinite search to a manageable one. We've gone from potentially looking at an endless range of numbers to a specific set of candidates. It's like having a map that guides us to the possible locations of buried treasure, instead of just wandering aimlessly. This application of the Rational Root Theorem isn't just a mathematical exercise; it's a demonstration of how theoretical tools can provide practical solutions, making complex problems solvable.

Testing the Potential Roots: Time to Put on Our Lab Coats

Alright, we've got our list of potential rational roots, thanks to the Rational Root Theorem. Now comes the slightly tedious but crucial part: testing each of these candidates. We need to see which ones actually make f(x) = 0. There are a couple of ways we can do this. One method is direct substitution: we plug each potential root into the function and see if the result is zero. For example, let's try x = 2: f(2) = 2(2)³ - 19(2)² + 57(2) - 54 = 16 - 76 + 114 - 54 = 0. Bingo! x = 2 is a rational root. But what if we want a more efficient way to test these roots? That's where synthetic division comes in handy. Synthetic division is a streamlined method for dividing a polynomial by a linear factor (x - c), where c is our potential root. If the remainder is zero, then c is indeed a root. Let's use synthetic division to test x = 3: We set up the synthetic division table with the coefficients of our polynomial (2, -19, 57, -54) and the potential root (3). Performing the synthetic division, we get a remainder of 0. This confirms that x = 3 is another rational root! Now, we could go through the entire list, testing each candidate one by one. However, there's a clever shortcut we can use. Remember, after performing synthetic division, the numbers in the bottom row (excluding the remainder) are the coefficients of the quotient polynomial. In our case with x = 3, the quotient is 2x² - 13x + 18. If we can factor this quadratic, we can find the remaining roots much faster! This step highlights the interconnectedness of different mathematical concepts. What started as a quest for rational roots has led us to factoring a quadratic, demonstrating the elegance and efficiency of mathematical problem-solving. By combining the Rational Root Theorem with synthetic division and factoring techniques, we're not just finding roots; we're building a deeper understanding of polynomial functions and their behavior.

Factoring the Quadratic and Finding the Final Root

Okay, we've used synthetic division and found that x = 3 is a root, leaving us with the quadratic 2x² - 13x + 18. Now, let's factor this bad boy! Factoring a quadratic involves finding two binomials that multiply together to give us the original quadratic. There are several techniques for factoring, but for this one, we can look for two numbers that multiply to (2 * 18) = 36 and add up to -13. After a bit of thought, we can see that -4 and -9 fit the bill. So, we can rewrite the middle term (-13x) as -4x - 9x and then factor by grouping: 2x² - 13x + 18 = 2x² - 4x - 9x + 18. Now, we group the terms: (2x² - 4x) + (-9x + 18). Factor out the greatest common factor from each group: 2x(x - 2) - 9(x - 2). Notice that we now have a common factor of (x - 2). Factor that out: (x - 2)(2x - 9). Awesome! We've successfully factored the quadratic. Now, to find the roots, we set each factor equal to zero: x - 2 = 0 => x = 2. 2x - 9 = 0 => x = 9/2. So, we've found two more roots: x = 2 and x = 9/2. Notice that x = 2 was actually one of the roots we found earlier when testing potential rational roots. This is a nice confirmation that our methods are consistent. We now have a complete list of the roots of the quadratic, which, combined with the root we found earlier (x = 3), gives us all the roots of the original cubic function. This process of factoring the quadratic is a powerful demonstration of how algebraic techniques can simplify complex problems. By breaking down the quadratic into its factors, we were able to easily identify the remaining roots, highlighting the elegance and efficiency of mathematical methods. The journey from the initial cubic function to the factored quadratic showcases the interconnectedness of different algebraic concepts and techniques.

The Grand Finale: Counting the Rational Roots

Alright guys, we've done the hard work! We used the Rational Root Theorem to narrow down the possibilities, employed synthetic division to test potential roots, and even factored a quadratic to find the remaining ones. Now, let's gather our findings and answer the original question: How many roots of f(x) are rational numbers? We found the following roots for f(x) = 2x³ - 19x² + 57x - 54: x = 2. x = 3. x = 9/2. Looking at these roots, we can see that all three of them are rational numbers! Remember, rational numbers can be expressed as a fraction p/q, where p and q are integers. 2 can be written as 2/1, 3 can be written as 3/1, and 9/2 is already in fraction form. So, there you have it! The function f(x) = 2x³ - 19x² + 57x - 54 has three rational roots. This final step is a testament to the power of systematic problem-solving in mathematics. By carefully applying the Rational Root Theorem, synthetic division, and factoring techniques, we were able to not only find the roots of the function but also confidently classify them as rational numbers. This process demonstrates how a combination of theoretical knowledge and practical application can lead to a complete and satisfying solution. The journey from the initial cubic function to the final count of rational roots is a celebration of mathematical reasoning and the joy of discovery.

In conclusion, by applying the Rational Root Theorem, employing synthetic division, and factoring the resulting quadratic, we successfully identified all the roots of the given cubic function f(x) = 2x³ - 19x² + 57x - 54. Our analysis revealed that the function has three rational roots: 2, 3, and 9/2. This comprehensive approach not only answers the question but also illustrates the interconnectedness of various mathematical concepts and techniques in solving polynomial equations.