Finding Values Of A And B With Given Factors
Introduction
In this article, we will delve into the process of finding the values of a and b in a polynomial expression, given that certain binomials are its factors. This is a common problem in algebra that combines polynomial factorization, the factor theorem, and equation solving. Specifically, we will tackle the problem of determining the values of a and b such that the expression (2x³ + ax² + x + b) has (x + 2) and (2x - 1) as factors. Understanding how to solve these types of problems is crucial for students studying algebra and for anyone interested in the deeper applications of polynomial equations.
The method we will use involves applying the factor theorem, which states that if (x - c) is a factor of a polynomial P(x), then P(c) = 0. By using this theorem, we can set up a system of equations based on the given factors and solve for the unknown coefficients a and b. This approach not only helps in finding the specific values but also reinforces the understanding of the relationship between polynomial roots and factors.
This article will provide a step-by-step solution, ensuring clarity and a thorough understanding of each stage. We will begin by explaining the underlying principles and then move on to the practical application of these concepts. By the end of this guide, you should be equipped to handle similar problems with confidence and precision. Let’s explore how we can unravel the mystery behind polynomial factorization and coefficient determination.
Understanding the Factor Theorem
Before we dive into the specifics of our problem, it's essential to grasp the factor theorem thoroughly. The factor theorem is a fundamental concept in algebra that connects the roots of a polynomial with its factors. In simple terms, it states that a polynomial P(x) has a factor (x - c) if and only if P(c) = 0. This theorem is a cornerstone in polynomial algebra and is invaluable for solving problems related to polynomial factorization and finding roots.
To illustrate, let’s consider a polynomial P(x). If we substitute a value c into the polynomial and find that P(c) equals zero, then we can conclude that (x - c) is indeed a factor of P(x). Conversely, if (x - c) is a factor of P(x), then substituting c into P(x) will result in zero. This bidirectional relationship is what makes the factor theorem so powerful.
The factor theorem is not just a theoretical concept; it has practical applications in various areas of mathematics and engineering. For example, it is used in the design of control systems, in signal processing, and in the analysis of circuits. Understanding and applying the factor theorem allows mathematicians and engineers to simplify complex problems, making calculations and analyses more manageable.
In the context of our problem, the factor theorem allows us to transform the information about the factors of the polynomial into equations involving the coefficients a and b. By setting the polynomial equal to zero when x is equal to the roots of the factors, we can create a system of equations that we can then solve to find the values of a and b. This is the core strategy we will employ to tackle the problem at hand.
Problem Statement and Setup
Our primary task is to find the values of the unknown coefficients a and b in the cubic polynomial P(x) = 2x³ + ax² + x + b. We are given that (x + 2) and (2x - 1) are factors of this polynomial. This information is crucial because it allows us to use the factor theorem to set up a system of equations that we can then solve to find the values of a and b.
The first step in solving this problem is to recognize that if (x + 2) is a factor of P(x), then according to the factor theorem, P(-2) must equal zero. Similarly, if (2x - 1) is a factor, then x = 1/2 is a root of the polynomial, meaning P(1/2) must also equal zero. This gives us two equations based on the factor theorem, which we can use to solve for our two unknowns, a and b.
To set up these equations, we will substitute x = -2 and x = 1/2 into the polynomial P(x) and set the results equal to zero. This process transforms the problem from one of polynomial factorization to one of solving a system of linear equations. The ability to make this transformation is a key skill in algebra, allowing us to apply familiar techniques to solve seemingly complex problems.
In the following sections, we will perform these substitutions and solve the resulting system of equations. This will lead us to the values of a and b that satisfy the conditions of the problem. Understanding how to set up the problem correctly is often the most challenging part, but once we have our equations, the solution becomes much more straightforward. Let's proceed to the next step and put these principles into action.
Applying the Factor Theorem
Now that we have a clear understanding of the problem and the factor theorem, we can proceed with the application of the theorem to our specific case. We have the polynomial P(x) = 2x³ + ax² + x + b, and we know that (x + 2) and (2x - 1) are its factors. This knowledge allows us to create two equations by substituting the roots of these factors into the polynomial and setting the result equal to zero.
First, let's consider the factor (x + 2). The root corresponding to this factor is x = -2. Substituting this value into P(x), we get:
P(-2) = 2(-2)³ + a(-2)² + (-2) + b = 0
Simplifying this equation, we have:
2(-8) + 4a - 2 + b = 0
-16 + 4a - 2 + b = 0
4a + b = 18 (Equation 1)
Next, we consider the factor (2x - 1). The root corresponding to this factor is x = 1/2. Substituting this value into P(x), we get:
P(1/2) = 2(1/2)³ + a(1/2)² + (1/2) + b = 0
Simplifying this equation, we have:
2(1/8) + a(1/4) + 1/2 + b = 0
1/4 + a/4 + 1/2 + b = 0
Multiplying the entire equation by 4 to eliminate fractions, we get:
1 + a + 2 + 4b = 0
a + 4b = -3 (Equation 2)
We now have a system of two linear equations with two variables, a and b. These equations are:
- 4a + b = 18
- a + 4b = -3
In the next section, we will solve this system of equations to find the values of a and b. The process of setting up these equations by applying the factor theorem is a crucial step in solving the problem, and we are now well-positioned to find the solution.
Solving the System of Equations
With the two equations we derived from the factor theorem, we now have a system of linear equations that we can solve to find the values of a and b. The equations are:
- 4a + b = 18
- a + 4b = -3
There are several methods to solve a system of linear equations, including substitution, elimination, and matrix methods. For this problem, we will use the elimination method, as it is efficient and straightforward.
To use the elimination method, we need to manipulate the equations so that the coefficients of one of the variables are the same (or opposites) in both equations. Let's eliminate the variable a. To do this, we can multiply the second equation by 4, which will make the coefficient of a in both equations equal.
Multiplying Equation 2 by 4, we get:
4(a + 4b) = 4(-3)
4a + 16b = -12 (Equation 3)
Now we have two equations:
- 4a + b = 18
- 4a + 16b = -12
To eliminate a, we can subtract Equation 3 from Equation 1:
(4a + b) - (4a + 16b) = 18 - (-12)
4a + b - 4a - 16b = 18 + 12
-15b = 30
Dividing both sides by -15, we find the value of b:
b = -2
Now that we have the value of b, we can substitute it back into either Equation 1 or Equation 2 to find the value of a. Let's use Equation 2:
a + 4(-2) = -3
a - 8 = -3
Adding 8 to both sides, we find the value of a:
a = 5
Thus, we have found that a = 5 and b = -2. These are the values that make (x + 2) and (2x - 1) factors of the polynomial 2x³ + ax² + x + b.
In the next section, we will verify our solution to ensure that it is correct. Verification is a crucial step in problem-solving, as it helps to catch any potential errors and ensures the accuracy of the final answer.
Verification of the Solution
After finding the values of a and b, it is essential to verify our solution to ensure its correctness. This step helps to confirm that our calculations are accurate and that the values we found satisfy the original problem conditions. We found that a = 5 and b = -2, so we will substitute these values back into the original polynomial and check if (x + 2) and (2x - 1) are indeed factors.
Our polynomial with the substituted values is:
P(x) = 2x³ + 5x² + x - 2
To verify that (x + 2) is a factor, we substitute x = -2 into P(x):
P(-2) = 2(-2)³ + 5(-2)² + (-2) - 2
P(-2) = 2(-8) + 5(4) - 2 - 2
P(-2) = -16 + 20 - 2 - 2
P(-2) = 0
Since P(-2) = 0, we confirm that (x + 2) is a factor of P(x).
Next, to verify that (2x - 1) is a factor, we substitute x = 1/2 into P(x):
P(1/2) = 2(1/2)³ + 5(1/2)² + (1/2) - 2
P(1/2) = 2(1/8) + 5(1/4) + 1/2 - 2
P(1/2) = 1/4 + 5/4 + 1/2 - 2
To combine these terms, we need a common denominator, which is 4:
P(1/2) = 1/4 + 5/4 + 2/4 - 8/4
P(1/2) = (1 + 5 + 2 - 8)/4
P(1/2) = 0/4
P(1/2) = 0
Since P(1/2) = 0, we confirm that (2x - 1) is also a factor of P(x).
Our verification process confirms that the values a = 5 and b = -2 are correct. This thorough check gives us confidence in our solution and demonstrates the importance of verification in mathematical problem-solving. By substituting the found values back into the original equation and confirming that the factors hold true, we ensure the accuracy of our results.
Conclusion
In this article, we successfully found the values of a and b such that the polynomial 2x³ + ax² + x + b has (x + 2) and (2x - 1) as factors. We achieved this by applying the factor theorem, which allowed us to set up a system of linear equations. This method demonstrates a powerful connection between the roots of a polynomial and its factors, providing a systematic way to solve such problems.
We began by understanding the factor theorem, which states that if (x - c) is a factor of a polynomial P(x), then P(c) = 0. This fundamental concept is the cornerstone of our solution. We then applied this theorem to our specific problem, setting up two equations by substituting the roots of the given factors into the polynomial. This transformed the problem from one of polynomial factorization to one of solving a system of linear equations.
Next, we solved the system of equations using the elimination method, which led us to find the values a = 5 and b = -2. Finally, we verified our solution by substituting these values back into the original polynomial and confirming that (x + 2) and (2x - 1) were indeed factors. This verification step is crucial in ensuring the accuracy of our solution.
This problem exemplifies the importance of understanding key algebraic concepts and how they can be applied to solve complex problems. By mastering the factor theorem and techniques for solving systems of equations, students can confidently tackle a wide range of algebraic challenges. The step-by-step approach outlined in this article provides a clear framework for solving similar problems in the future. Algebra is not just about formulas and equations; it is about understanding the relationships between mathematical concepts and applying them effectively to find solutions.
