Solving Polynomial Equations With Complex Roots

Hey guys! Today, we're diving into the fascinating world of polynomial equations, specifically those with complex roots. We'll be tackling a problem where we're given a cubic equation and some of its roots, and our mission is to find the remaining root. This is a classic problem that combines algebra and complex number theory, so buckle up and let's get started!

Understanding the Problem

Before we jump into the solution, let's break down the problem. We're given the polynomial equation:

x³ - 4x² + 2x + 10 = x² - 5x - 3

This is a cubic equation, meaning it's a polynomial equation of degree 3. This tells us that the equation will have three roots (solutions), which may be real or complex. We're also given that two of the roots are complex numbers: 3 + 2i and 3 - 2i. Remember that complex roots always come in conjugate pairs if the polynomial has real coefficients, which is the case here. Our goal is to find the third root.

The complex roots provided, 3 + 2i and 3 - 2i, are complex conjugates. This is a crucial piece of information because it tells us that the coefficients of our polynomial are real numbers. This property is based on the Complex Conjugate Root Theorem, which states that if a polynomial with real coefficients has a complex root a + bi, then its conjugate a - bi is also a root. Understanding this theorem is key to solving this type of problem efficiently. When working with polynomials, especially in the context of complex roots, it's super important to recall fundamental theorems like the Complex Conjugate Root Theorem. This theorem not only confirms the structure of the roots but also guides our approach to finding the remaining roots. By identifying the complex conjugate pairs, we simplify the polynomial and make it easier to manage. Furthermore, by ensuring that our methodology aligns with these mathematical principles, we can confidently tackle polynomial equations that include complex numbers and gain a deeper understanding of algebraic structures.

Simplifying the Equation

First things first, let's simplify the equation by moving all the terms to one side:

x³ - 4x² + 2x + 10 - (x² - 5x - 3) = 0

Combining like terms, we get:

x³ - 5x² + 7x + 13 = 0

Now we have a standard cubic equation that we can work with. Remember, the goal here is to find the value(s) of x that make this equation true. We already know two of these values, thanks to the given complex roots.

The process of simplifying the equation is a fundamental step that should not be overlooked. By reorganizing and combining like terms, we transform the initial equation into a more manageable form, which is crucial for further analysis and problem-solving. The simplified polynomial, x³ - 5x² + 7x + 13 = 0, is now in a standard cubic form, making it easier to apply various algebraic techniques to find the roots. This step reduces the complexity of the equation, making it less intimidating and more approachable. Simplifying equations is not just about tidying up; it's a strategic move that can reveal underlying structures and relationships. In this instance, the simplified equation allows us to clearly see the coefficients and constants, which are essential for applying methods such as synthetic division or polynomial factorization. In essence, simplification prepares the equation for more advanced algebraic manipulations and sets the stage for efficiently determining the roots. By paying close attention to this step, we can prevent errors and streamline the solution process.

Using the Roots to Factor the Polynomial

Since we know two roots, 3 + 2i and 3 - 2i, we can use them to factor the polynomial. If r is a root of a polynomial, then (x - r) is a factor of the polynomial. So, we have two factors:

(x - (3 + 2i)) and (x - (3 - 2i))

Let's multiply these factors together to get a quadratic factor:

(x - (3 + 2i))(x - (3 - 2i)) = (x - 3 - 2i)(x - 3 + 2i)

This might look a bit intimidating, but we can use the difference of squares pattern: (a - b)(a + b) = a² - b². Let a = (x - 3) and b = 2i. Then,

(x - 3 - 2i)(x - 3 + 2i) = (x - 3)² - (2i)²

Expanding this, we get:

(x - 3)² - (2i)² = x² - 6x + 9 - (4i²) = x² - 6x + 9 - (-4) = x² - 6x + 13

So, we have a quadratic factor of x² - 6x + 13.

The process of using the roots to factor the polynomial is a crucial step in solving polynomial equations. This technique stems directly from the Factor Theorem, which states that if r is a root of a polynomial P(x), then (x - r) is a factor of P(x). By applying this theorem, we transform the knowledge of the roots into a tangible algebraic component, namely the factors of the polynomial. In this case, the given complex roots 3 + 2i and 3 - 2i lead us to the factors (x - (3 + 2i)) and (x - (3 - 2i)). Multiplying these factors together is a strategic move because it consolidates the complex roots into a single quadratic factor with real coefficients, which simplifies further calculations. The resulting quadratic factor, x² - 6x + 13, is a pivotal element that allows us to reduce the complexity of the original cubic equation. This method effectively bridges the gap between abstract root values and concrete algebraic expressions, illustrating the power of factorization in polynomial manipulation. By employing this technique, we not only make the equation easier to handle but also gain a deeper insight into the structure and properties of polynomials.

Finding the Remaining Root

Now that we have a quadratic factor, we can divide the original cubic polynomial by this factor to find the remaining linear factor, which will give us the other root. We can use polynomial long division or synthetic division. Let's use polynomial long division:

 x + 1
 x² - 6x + 13 | x³ - 5x² + 7x + 13
 - (x³ - 6x² + 13x)
 ------------------
 x² - 6x + 13
 - (x² - 6x + 13)
 ------------------
 0

So, the quotient is x + 1. This means that our cubic polynomial can be factored as:

x³ - 5x² + 7x + 13 = (x² - 6x + 13)(x + 1)

The remaining factor is (x + 1), which gives us the other root:

x + 1 = 0

x = -1

Therefore, the other root of the polynomial equation is -1.

The process of finding the remaining root involves leveraging the quadratic factor we previously determined through complex conjugate roots and employing polynomial division. This step is a quintessential application of the Polynomial Remainder Theorem, which essentially states that if you divide a polynomial P(x) by (x - c) and the remainder is zero, then c is a root of P(x). By dividing the original cubic polynomial, x³ - 5x² + 7x + 13, by the quadratic factor x² - 6x + 13, we methodically reduce the cubic to a simpler, manageable linear factor. The successful long division, which yields the quotient x + 1 and a remainder of zero, confirms that x² - 6x + 13 is indeed a factor, as expected. This outcome directly leads us to the remaining factor (x + 1), which, when set equal to zero, unveils the real root x = -1. This technique is not merely about algebraic manipulation; it demonstrates the elegant way in which factorization and division can systematically uncover the roots of a polynomial. By applying the Polynomial Remainder Theorem, we not only find the root but also validate our factorization, reinforcing the interconnectedness of roots and factors in polynomial equations.

Final Answer

The polynomial equation x³ - 4x² + 2x + 10 = x² - 5x - 3 has complex roots 3 + 2i and 3 - 2i, and the other root is -1.

Key Takeaways

• Complex roots of polynomials with real coefficients always come in conjugate pairs.
• If r is a root of a polynomial, then (x - r) is a factor of the polynomial.
• Polynomial long division or synthetic division can be used to find remaining factors and roots.
• The Factor Theorem and Complex Conjugate Root Theorem are powerful tools for solving polynomial equations.

Why This Matters

Understanding how to solve polynomial equations, especially those with complex roots, is crucial in many areas of mathematics and engineering. Polynomials are used to model a wide variety of phenomena, from the motion of projectiles to the behavior of electrical circuits. Knowing how to find the roots of a polynomial allows us to analyze the behavior of these systems and make predictions about their future states. For example, in control systems engineering, the roots of the characteristic polynomial determine the stability of a system. Complex roots can indicate oscillations or instability, while real roots indicate stable behavior. Similarly, in signal processing, the roots of a polynomial can be used to design filters that selectively amplify or attenuate certain frequencies.

Polynomial equations, particularly those with complex roots, are fundamental in various fields, making their understanding and resolution crucial. In electrical engineering, the analysis of AC circuits often involves solving polynomial equations with complex roots, as the impedance of circuit elements (resistors, capacitors, and inductors) can be represented using complex numbers. The solutions to these equations are essential for determining the circuit's stability and response to different inputs. In mechanical engineering, vibrational analysis frequently uses polynomials to model the behavior of mechanical systems. Complex roots in this context can represent damped oscillations, providing critical information for designing systems that minimize unwanted vibrations. Furthermore, in control systems engineering, the characteristic equation of a system, which is a polynomial, plays a vital role in determining the system's stability. The roots of this polynomial indicate whether the system will converge to a stable state or oscillate, influencing the design of control strategies. Understanding and solving these polynomial equations, therefore, underpins the ability to create stable and efficient systems.

In physics, complex roots arise in quantum mechanics when solving the Schrödinger equation for certain potentials. The complex solutions represent quantum states with specific energy levels, and understanding these roots is essential for predicting the behavior of quantum systems. In computer graphics, polynomial equations are used to create curves and surfaces, such as Bézier curves and splines, which are fundamental in generating realistic images and animations. The roots of these polynomials determine the shape and characteristics of the curves, influencing the aesthetic quality of the visuals. Moreover, in cryptography, polynomial equations are used in the design of encryption algorithms. The security of many cryptographic systems relies on the difficulty of finding the roots of certain polynomials, ensuring the protection of sensitive information. In each of these applications, the ability to solve polynomial equations, especially those with complex roots, is not just an abstract mathematical skill but a practical necessity for advancing innovation and solving real-world problems.

Practice Makes Perfect

To really master this topic, try solving similar problems on your own. Look for practice questions in your textbook or online. The more you practice, the more comfortable you'll become with the techniques involved. Remember, math is like a muscle – the more you use it, the stronger it gets!

So, there you have it! We've successfully navigated a cubic equation with complex roots and found the missing piece of the puzzle. Keep practicing, and you'll be a polynomial pro in no time!