Hey guys! Today, we're diving into the exciting world of polynomial functions and tackling the challenge of finding the zeros of a cubic function. Specifically, we're going to find the zeros of the function f(x) = x³ - 10x² + 44x - 69. This might seem daunting at first, but don't worry, we'll break it down step by step and make it super easy to understand. Zeros, also known as roots, are the values of x that make the function equal to zero. These are the points where the graph of the function crosses the x-axis. For a cubic function like ours, we can expect to find up to three zeros, which can be real or complex numbers. Understanding how to find these zeros is crucial in various fields, including engineering, physics, and computer science. So, let's get started and unlock the secrets of this cubic function!
Understanding Polynomial Functions
Before we jump into the solution, let's take a moment to understand what polynomial functions are and why finding their zeros is so important. A polynomial function is a function that can be expressed in the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ, aₙ₋₁, ..., a₁, a₀ are constants called coefficients, and n is a non-negative integer called the degree of the polynomial. In our case, f(x) = x³ - 10x² + 44x - 69 is a cubic polynomial because the highest power of x is 3. The zeros of a polynomial function are the values of x for which f(x) = 0. These zeros are also known as the roots of the polynomial equation. Finding the zeros of a polynomial is a fundamental problem in algebra, with applications in many different areas. For example, in engineering, zeros can represent the resonant frequencies of a system. In physics, they can represent equilibrium points. And in computer science, they can be used in root-finding algorithms. The process of finding zeros can sometimes be straightforward, especially for linear and quadratic functions. However, for higher-degree polynomials like cubics, it can become more challenging, often requiring techniques like the Rational Root Theorem, synthetic division, or numerical methods. Understanding the nature of polynomials and their zeros is essential for anyone working with mathematical models and real-world applications.
The Importance of Finding Zeros
Finding the zeros of a function isn't just a mathematical exercise; it's a crucial skill with wide-ranging applications in various fields. Think of zeros as the key points where a function intersects the x-axis. These points often represent significant values or states in real-world scenarios. In engineering, for instance, finding the zeros of a system's transfer function can help determine its stability. Imagine designing a bridge; you'd need to know the points where the structure is most vulnerable to stress – those are often related to the zeros of the system's equations. In physics, zeros can represent equilibrium points, where forces balance each other out. Consider a pendulum swinging back and forth; the points where it momentarily stops before changing direction are related to the zeros of the equation describing its motion. Even in economics, zeros can represent break-even points, where costs equal revenue. So, understanding how to find these zeros allows economists to predict when a business will start making a profit. The ability to find zeros also unlocks more advanced mathematical techniques. Knowing the zeros of a polynomial, for example, allows us to factor it, which can simplify complex equations and make them easier to solve. This is particularly useful in calculus, where finding zeros is often a necessary step in optimization problems or finding areas under curves. In essence, finding zeros is a fundamental tool in a mathematician's toolkit, enabling us to solve problems, model real-world phenomena, and make informed decisions in a variety of disciplines. So, mastering this skill opens doors to a deeper understanding of the world around us.
Methods to Find Zeros of Polynomials
Alright, so how do we actually find these elusive zeros? There are several methods we can use, depending on the degree and complexity of the polynomial. For linear and quadratic functions, things are pretty straightforward. Linear functions (degree 1) have a simple form, f(x) = mx + b, and their zero can be found by setting f(x) = 0 and solving for x. Quadratic functions (degree 2), of the form f(x) = ax² + bx + c, can be solved using the quadratic formula, factoring, or completing the square. But when we move on to cubic functions (degree 3) and higher, things get a bit more interesting. One powerful tool in our arsenal is the Rational Root Theorem. This theorem helps us narrow down the possible rational roots (zeros that can be expressed as a fraction) of a polynomial. It states that if a polynomial with integer coefficients has a rational root p/q (where p and q are integers with no common factors), then p must be a factor of the constant term and q must be a factor of the leading coefficient. This gives us a limited set of potential rational roots to test. Once we have a potential root, we can use synthetic division to test it. Synthetic division is a shortcut method for dividing a polynomial by a linear factor (x - r), where r is the potential root. If the remainder after synthetic division is zero, then r is indeed a root of the polynomial, and we've successfully factored out a linear term. After factoring out a linear term, we are left with a polynomial of lower degree, which is easier to handle. For example, if we start with a cubic and find one rational root, we're left with a quadratic, which we can solve using the methods mentioned earlier. In some cases, however, the polynomial may not have any rational roots. In such situations, we might need to resort to numerical methods, such as the Newton-Raphson method, or use computer software to approximate the roots. These methods provide numerical approximations rather than exact solutions. So, depending on the polynomial we're dealing with, we have a variety of tools and techniques at our disposal to find its zeros.
Rational Root Theorem
The Rational Root Theorem is like a detective's magnifying glass for polynomial equations, helping us zoom in on potential rational zeros. This theorem is a powerful tool when dealing with polynomials that have integer coefficients, as it provides a systematic way to narrow down the possible rational roots. Let's break down how it works. Imagine we have a polynomial equation like aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0, where all the coefficients (aₙ, aₙ₋₁, ..., a₁, a₀) are integers. The Rational Root Theorem tells us that if this polynomial has any rational roots (roots that can be expressed as a fraction p/q), then p must be a factor of the constant term (a₀) and q must be a factor of the leading coefficient (aₙ). In simpler terms, we look at the factors of the constant term (the number without any x attached) and the factors of the leading coefficient (the number in front of the highest power of x). We then form all possible fractions by dividing the factors of the constant term by the factors of the leading coefficient. These fractions (both positive and negative) are our potential rational roots. The theorem doesn't guarantee that any of these potential roots are actual roots, but it gives us a finite list to test, which is a huge advantage compared to blindly guessing. For example, let's say we have the polynomial 2x³ - 5x² + 4x - 1 = 0. The constant term is -1, and its factors are ±1. The leading coefficient is 2, and its factors are ±1 and ±2. So, the possible rational roots are ±1/1 and ±1/2, which simplifies to ±1 and ±1/2. We now have just four values to test, rather than an infinite number of possibilities. To test these values, we can use synthetic division or direct substitution. The Rational Root Theorem is especially useful for polynomials of degree 3 or higher, where factoring can be challenging. It's a cornerstone technique in polynomial algebra, making the search for zeros a much more manageable task.
Synthetic Division
Once we've identified potential rational roots using the Rational Root Theorem, the next step is to test them, and that's where synthetic division comes into play. Think of synthetic division as a streamlined, efficient way to divide a polynomial by a linear factor, like (x - r). It's a shortcut method that simplifies the division process, making it much faster and less prone to errors than traditional long division. The key idea behind synthetic division is to focus on the coefficients of the polynomial and the potential root, rather than dealing with the x terms explicitly. Let's walk through the process. First, we write down the coefficients of the polynomial in a row, making sure to include zeros as placeholders for any missing terms. For example, if we have the polynomial x³ - 2x + 1, we would write down 1 (for x³), 0 (for the missing x² term), -2 (for x), and 1 (the constant term). Next, we write the potential root, r, to the left of the coefficients. Now, the magic begins! We bring down the first coefficient to the bottom row. Then, we multiply this coefficient by r and write the result under the next coefficient. We add the two numbers in that column and write the sum in the bottom row. We repeat this process – multiply the last number in the bottom row by r and write the result under the next coefficient, then add – until we've reached the last coefficient. The last number in the bottom row is the remainder. If the remainder is zero, that means (x - r) is a factor of the polynomial, and r is a root! The other numbers in the bottom row are the coefficients of the quotient polynomial, which is one degree lower than the original polynomial. So, synthetic division not only tells us whether a potential root is a true root, but it also helps us factor the polynomial. This is incredibly useful because once we've found one root, we can reduce the polynomial to a lower degree, making it easier to find the remaining roots. Synthetic division is a workhorse in polynomial algebra, saving us time and effort in the quest for polynomial zeros.
Applying the Methods to Our Function
Okay, let's get back to our original function, f(x) = x³ - 10x² + 44x - 69. We're going to use the methods we've discussed to find its zeros. First up, the Rational Root Theorem. The constant term is -69, and its factors are ±1, ±3, ±23, and ±69. The leading coefficient is 1, and its factors are simply ±1. So, the possible rational roots are ±1, ±3, ±23, and ±69. That's quite a few possibilities, but it's still a manageable list compared to an infinite number of potential roots! Now, let's put these potential roots to the test using synthetic division. We'll start with the simplest one, 1. After performing synthetic division with 1, we find that the remainder is not zero, so 1 is not a root. Let's try -1. Again, the remainder is not zero, so -1 is not a root either. Next up, let's try 3. When we perform synthetic division with 3, we get a remainder of zero! This is fantastic news – it means 3 is a root of our function. The synthetic division also gives us the quotient polynomial, which is x² - 7x + 23. So, we've factored our cubic function as (x - 3)(x² - 7x + 23). Now, we need to find the zeros of the quadratic factor, x² - 7x + 23. Since this quadratic doesn't seem to factor easily, we'll use the quadratic formula. The quadratic formula is a reliable tool for finding the roots of any quadratic equation of the form ax² + bx + c = 0. It states that x = (-b ± √(b² - 4ac)) / 2a. In our case, a = 1, b = -7, and c = 23. Plugging these values into the quadratic formula, we get x = (7 ± √((-7)² - 4 * 1 * 23)) / (2 * 1), which simplifies to x = (7 ± √(-43)) / 2. Since we have a negative number under the square root, we know that the roots will be complex numbers. We can rewrite √(-43) as √43 * i, where i is the imaginary unit (√-1). So, the roots of the quadratic factor are (7 ± √43 i) / 2. Therefore, the zeros of the original function f(x) = x³ - 10x² + 44x - 69 are 3, (7 + √43 i) / 2, and (7 - √43 i) / 2. We've successfully found all three zeros, one real and two complex! This demonstrates how we can combine the Rational Root Theorem, synthetic division, and the quadratic formula to solve cubic equations and find their zeros.
Detailed Steps
Let's break down the solution into detailed steps, so you can follow along and understand exactly how we found the zeros of f(x) = x³ - 10x² + 44x - 69. This step-by-step approach will make the process crystal clear. Step 1: Apply the Rational Root Theorem. First, we identify the constant term, which is -69, and the leading coefficient, which is 1. The factors of -69 are ±1, ±3, ±23, and ±69. The factors of 1 are ±1. According to the Rational Root Theorem, the possible rational roots are the fractions formed by dividing the factors of the constant term by the factors of the leading coefficient. This gives us the list: ±1, ±3, ±23, and ±69. Step 2: Test potential roots using synthetic division. We'll start by testing 1. Setting up the synthetic division, we get:
1 | 1 -10 44 -69
| 1 -9 35
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1 -9 35 -34
The remainder is -34, which is not zero, so 1 is not a root. Next, let's try -1:
-1 | 1 -10 44 -69
| -1 11 -55
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1 -11 55 -124
The remainder is -124, so -1 is not a root. Now, let's try 3:
3 | 1 -10 44 -69
| 3 -21 69
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1 -7 23 0
Bingo! The remainder is 0, so 3 is a root. The quotient is x² - 7x + 23. Step 3: Solve the quadratic equation. We now have the factored form: (x - 3)(x² - 7x + 23) = 0. We need to find the zeros of the quadratic factor, x² - 7x + 23. Since it doesn't factor easily, we'll use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. In our case, a = 1, b = -7, and c = 23. Plugging these values in, we get:
x = (7 ± √((-7)² - 4 * 1 * 23)) / (2 * 1) x = (7 ± √(49 - 92)) / 2 x = (7 ± √(-43)) / 2 x = (7 ± √43 i) / 2 Step 4: State the zeros. The zeros of the function f(x) = x³ - 10x² + 44x - 69 are 3, (7 + √43 i) / 2, and (7 - √43 i) / 2. By following these detailed steps, you can confidently find the zeros of similar cubic functions. Remember, practice makes perfect, so try applying these techniques to other examples!
Solutions
Alright, let's recap the solutions we found for the zeros of the function f(x) = x³ - 10x² + 44x - 69. After applying the Rational Root Theorem, synthetic division, and the quadratic formula, we successfully identified all three zeros. We found one real zero and two complex zeros, which is typical for a cubic function. The real zero is x = 3. This means that the graph of the function crosses the x-axis at the point (3, 0). The complex zeros are x = (7 + √43 i) / 2 and x = (7 - √43 i) / 2. These zeros are a conjugate pair, meaning they have the same real part but opposite imaginary parts. This is a common occurrence when dealing with polynomial equations with real coefficients. Complex zeros don't show up as x-intercepts on the graph of the function, but they are still important solutions to the equation f(x) = 0. So, to summarize, the zeros of f(x) = x³ - 10x² + 44x - 69 are:
- Real zero: x = 3
- Complex zeros: x = (7 + √43 i) / 2 and x = (7 - √43 i) / 2
We've tackled this problem using a combination of algebraic techniques, demonstrating the power of the Rational Root Theorem, synthetic division, and the quadratic formula. Finding zeros of functions is a fundamental skill in mathematics, with applications in various fields, from engineering to physics to computer science. By mastering these techniques, you'll be well-equipped to solve a wide range of problems and gain a deeper understanding of the mathematical world.
Conclusion
In conclusion, we've successfully navigated the process of finding the zeros of the cubic function f(x) = x³ - 10x² + 44x - 69. We started by understanding the importance of finding zeros and the various methods available to us. We then applied the Rational Root Theorem to narrow down the possible rational roots, followed by synthetic division to test these potential roots. This led us to identify one real root, x = 3. With one root in hand, we factored the cubic function into a linear factor and a quadratic factor. To find the zeros of the quadratic factor, we employed the quadratic formula, which revealed two complex roots: (7 + √43 i) / 2 and (7 - √43 i) / 2. Therefore, the complete set of zeros for the function f(x) consists of one real root and two complex roots. This exercise demonstrates the power and versatility of algebraic techniques in solving polynomial equations. The combination of the Rational Root Theorem, synthetic division, and the quadratic formula provides a systematic approach to finding zeros, even for higher-degree polynomials. Finding zeros is a fundamental skill in mathematics, with applications in numerous fields, including engineering, physics, and computer science. By mastering these techniques, you can solve a wide variety of problems and deepen your understanding of mathematical concepts. Remember, practice is key to success. Try applying these methods to other polynomial functions to solidify your understanding and build your problem-solving skills. Keep exploring the world of mathematics, and you'll discover its beauty and power in solving real-world challenges!