Solving X⁴ + 9x³ + 12x² - 11x + 21 = 0 Find The Real Solutions
Hey everyone! Today, we're diving into a fun mathematical journey to find the real solutions of the polynomial equation:
x⁴ + 9x³ + 12x² - 11x + 21 = 0
Polynomial equations can sometimes look intimidating, but don't worry, we'll break it down step by step. We will explore different techniques for solving such equations, ensuring you grasp the concepts thoroughly. Let's get started!
Understanding the Problem
Before we jump into solving, let's understand what we're dealing with. We have a quartic equation (degree 4), which means it can have up to four solutions. Our goal is to find the real number solutions, if any exist. Real solutions are the values of x that, when plugged into the equation, make the equation true and are not imaginary numbers. To effectively tackle this problem, we’ll need to employ a mix of algebraic techniques and analytical thinking. It’s crucial to have a solid grasp of polynomial functions and their behavior to navigate through the solution process successfully. This involves understanding concepts like the degree of the polynomial, leading coefficients, and how these factors influence the shape and roots of the function.
Initial Observations and Considerations
First, let's make some initial observations. The equation is:
x⁴ + 9x³ + 12x² - 11x + 21 = 0
Notice that all the coefficients are real numbers. This is important because it means that if there are any complex roots, they will come in conjugate pairs. Also, the constant term is 21, which will be helpful when we consider potential rational roots using the Rational Root Theorem. The Rational Root Theorem provides a list of potential rational roots by considering the factors of the constant term (21) divided by the factors of the leading coefficient (1). This gives us a starting point for testing possible solutions and can significantly narrow down the search. Moreover, observing the signs of the coefficients can give us hints about the nature of the roots. For instance, Descartes' Rule of Signs can help predict the number of positive and negative real roots.
Methods to Find Real Solutions
There are several methods we can use to find the real solutions of this equation. Let's explore some of them:
1. Rational Root Theorem
The Rational Root Theorem states that if a polynomial equation with integer coefficients has rational roots, they must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our case:
- Constant term: 21 (factors: ±1, ±3, ±7, ±21)
- Leading coefficient: 1 (factors: ±1)
So, the possible rational roots are: ±1, ±3, ±7, ±21. Let's test these values by plugging them into the equation.
Testing these values is a critical step. We substitute each potential root into the original equation and check if the result is zero. This process, while sometimes tedious, can quickly identify rational roots if they exist. If we find a rational root, we can then use synthetic division or polynomial long division to reduce the degree of the polynomial, making it easier to find other roots. For instance, if x = -3 is a root, we can divide the polynomial by (x + 3) to obtain a cubic equation, which might be simpler to solve. This iterative process of finding roots and reducing the polynomial's degree is a common strategy in solving polynomial equations.
2. Descartes' Rule of Signs
Descartes' Rule of Signs helps us determine the possible number of positive and negative real roots of a polynomial equation. To find the possible number of positive real roots, we count the number of sign changes in the coefficients of the polynomial. For negative real roots, we substitute x with -x and count the sign changes.
Our equation is:
x⁴ + 9x³ + 12x² - 11x + 21 = 0
- Sign changes: + to +, + to +, + to -, - to + (2 sign changes)
So, there are either 2 or 0 positive real roots. Now, let's substitute x with -x:
(-x)⁴ + 9(-x)³ + 12(-x)² - 11(-x) + 21 = 0
x⁴ - 9x³ + 12x² + 11x + 21 = 0
- Sign changes: + to -, - to +, + to +, + to + (2 sign changes)
So, there are either 2 or 0 negative real roots. Descartes' Rule of Signs is a powerful tool for narrowing down the possibilities. It gives us a range for the number of positive and negative real roots, which can guide our search and prevent us from wasting time on fruitless attempts. However, it's important to remember that the rule only provides the possible number of roots; there might be fewer real roots than the rule suggests, especially if some roots are complex.
3. Numerical Methods and Graphing
When analytical methods don't easily yield solutions, numerical methods and graphing can be invaluable. Graphing the polynomial can give us a visual representation of the function's behavior, allowing us to estimate where the roots might be. Numerical methods, such as the Newton-Raphson method, use iterative algorithms to approximate the roots to a desired level of accuracy. These methods are particularly useful for equations that are difficult or impossible to solve algebraically. By using graphing tools or software, we can quickly sketch the curve of the polynomial function. The points where the graph intersects the x-axis represent the real roots of the equation. This visual approach can help us identify intervals where roots are likely to exist, making it easier to apply numerical methods or to refine our search using analytical techniques.
Applying the Methods
Let's start by testing the possible rational roots we found earlier (±1, ±3, ±7, ±21):
- For x = 1: 1 + 9 + 12 - 11 + 21 = 32 ≠ 0
- For x = -1: 1 - 9 + 12 + 11 + 21 = 36 ≠ 0
- For x = 3: 81 + 243 + 108 - 33 + 21 = 420 ≠ 0
- For x = -3: 81 - 243 + 108 + 33 + 21 = 0
We found a root! x = -3 is a solution. Now, we can use synthetic division or polynomial long division to divide the polynomial by (x + 3).
Polynomial Division
Performing polynomial division, we divide x⁴ + 9x³ + 12x² - 11x + 21 by (x + 3). This process allows us to reduce the quartic equation to a cubic equation, which is generally easier to handle. The result of the division gives us the quotient, which is a polynomial of degree three, and the remainder, which should be zero if (x + 3) is indeed a factor. This step is crucial because it simplifies the original problem into a more manageable form. By focusing on the cubic equation, we can apply similar techniques, such as the Rational Root Theorem or numerical methods, to find the remaining roots. The division process not only helps in finding roots but also provides a deeper understanding of the polynomial's structure and factorization.
The result is: x³ + 6x² - 6x + 7
So, we have:
(x + 3)(x³ + 6x² - 6x + 7) = 0
Now, we need to find the roots of the cubic equation x³ + 6x² - 6x + 7 = 0. Let’s apply the Rational Root Theorem again.
Rational Root Theorem for Cubic Equation
For the cubic equation x³ + 6x² - 6x + 7 = 0, the possible rational roots are the factors of 7 (±1, ±7). Let's test them:
- For x = -7: (-7)³ + 6(-7)² - 6(-7) + 7 = -343 + 294 + 42 + 7 = 0
We found another root! x = -7 is a solution. Now, divide the cubic equation by (x + 7).
Dividing the Cubic Polynomial
Dividing the cubic polynomial x³ + 6x² - 6x + 7 by (x + 7) using synthetic division or polynomial long division, we aim to reduce the equation to a quadratic form. This step is significant because quadratic equations have well-established methods for finding solutions, such as the quadratic formula. The result of this division will give us a quadratic equation, and we can then use the quadratic formula to determine its roots. This process demonstrates a systematic approach to solving higher-degree polynomial equations by breaking them down into simpler, more manageable forms. Each step builds upon the previous one, leading us closer to the complete set of solutions.
The result is: x² - x + 1
So, we have:
(x + 3)(x + 7)(x² - x + 1) = 0
Now, we need to solve the quadratic equation x² - x + 1 = 0. We can use the quadratic formula:
Solving the Quadratic Equation
To solve the quadratic equation x² - x + 1 = 0, we employ the quadratic formula, which is a general method for finding the roots of any quadratic equation of the form ax² + bx + c = 0. The formula is given by x = (-b ± √(b² - 4ac)) / (2a). In our case, a = 1, b = -1, and c = 1. Substituting these values into the formula allows us to find the roots of the equation. This method is particularly useful when the quadratic equation cannot be easily factored. By systematically applying the quadratic formula, we can determine the nature of the roots – whether they are real or complex – and find their exact values. This step is a critical part of the overall solution process, as it provides the final pieces needed to understand the complete solution set of the original polynomial equation.
x = (-(-1) ± √((-1)² - 4(1)(1))) / (2(1))
x = (1 ± √(1 - 4)) / 2
x = (1 ± √(-3)) / 2
x = (1 ± i√3) / 2
The quadratic equation has complex solutions:
x = (1 + i√3) / 2 and x = (1 - i√3) / 2
Final Solution
We found two real solutions and two complex solutions. The real solutions are x = -3 and x = -7.
Summarizing the Solution Set
In summary, by applying a combination of the Rational Root Theorem, Descartes' Rule of Signs, polynomial division, and the quadratic formula, we have successfully found all the solutions to the equation x⁴ + 9x³ + 12x² - 11x + 21 = 0. The real solutions are x = -3 and x = -7, while the complex solutions are x = (1 + i√3) / 2 and x = (1 - i√3) / 2. This comprehensive approach demonstrates the power of combining different mathematical techniques to tackle complex problems. Understanding and applying these methods not only helps in finding the solutions but also enhances our understanding of the underlying mathematical principles. The process highlights the importance of systematic problem-solving and the interconnectedness of various mathematical concepts.
Therefore, the solution set is {-7, -3}.
A. The solution set is {-7, -3}
I hope this detailed explanation helps you understand the process of finding real solutions for polynomial equations. Keep practicing, and you'll become a pro in no time!
