Finding The Other Factor Of 6x²-7x+2 A Step-by-Step Solution
As we delve into the fascinating world of algebra, understanding factorization is paramount. Factoring polynomials allows us to simplify complex expressions and solve equations, making it a fundamental skill in mathematics. In this comprehensive guide, we will explore the factorization of the quadratic expression 6x² - 7x + 2, given that one of its factors is (2x - 1). Our journey will involve a step-by-step approach, ensuring clarity and a deep understanding of the underlying concepts. We will dissect the problem, explore different methods of factorization, and ultimately unveil the other factor of this expression. The goal is not just to arrive at the correct answer but to empower you with the knowledge and confidence to tackle similar problems with ease. So, let's embark on this algebraic adventure together and unlock the secrets of factorization.
Understanding the Problem: Deconstructing 6x² - 7x + 2
Before we jump into the solution, let's take a moment to fully grasp the problem at hand. We are presented with the quadratic expression 6x² - 7x + 2, a polynomial of degree two. This expression is composed of three terms: a quadratic term (6x²), a linear term (-7x), and a constant term (2). The coefficients of these terms play a crucial role in the factorization process. We are also given that (2x - 1) is one of the factors of this expression. A factor, in this context, is an expression that divides evenly into the given polynomial. In other words, if we multiply (2x - 1) by another expression, we should obtain 6x² - 7x + 2. Our task is to find this other expression, the other factor, that when multiplied by (2x - 1), yields the original quadratic expression. This process is akin to reverse multiplication, where we are given the product and one of the multipliers, and we need to find the other multiplier. To achieve this, we will explore various factorization techniques and strategically apply them to this specific problem. Understanding the structure of the quadratic expression and the meaning of factors is the first crucial step in our quest to find the other factor.
Methods of Factorization: A Toolkit for Polynomial Decomposition
Factorization, the art of breaking down a polynomial into its constituent factors, is a cornerstone of algebraic manipulation. There are several methods at our disposal, each with its strengths and applicable scenarios. One of the most common techniques is the splitting the middle term method, which involves rewriting the middle term of the quadratic expression as a sum or difference of two terms, allowing us to factor by grouping. Another powerful method is the quadratic formula, which provides a direct way to find the roots of a quadratic equation, and from these roots, we can construct the factors. The trial and error method, while sometimes tedious, can be effective for simpler quadratic expressions, where we systematically try different combinations of factors until we find the correct one. Synthetic division can also be used to divide the quadratic expression by the known factor (2x - 1). In addition to these algebraic techniques, visual aids such as diagrams and area models can provide a geometric interpretation of factorization, making the process more intuitive. Each method offers a unique perspective on factorization, and the choice of method often depends on the specific problem and personal preference. In our quest to find the other factor of 6x² - 7x + 2, we will carefully consider these methods and select the most efficient one for this particular scenario.
Solving for the Other Factor: A Step-by-Step Approach
Now, let's put our factorization toolkit to work and find the other factor of 6x² - 7x + 2, given that (2x - 1) is one factor. We can approach this problem using a couple of different methods. One effective method is polynomial long division. We can divide 6x² - 7x + 2 by (2x - 1) to find the quotient, which will be the other factor. Alternatively, we can use a more intuitive approach based on the distributive property. We know that (2x - 1) multiplied by the other factor should give us 6x² - 7x + 2. Let's assume the other factor is of the form (ax + b), where a and b are constants that we need to determine. So, we have: (2x - 1)(ax + b) = 6x² - 7x + 2. Expanding the left side, we get: 2ax² + (2b - a)x - b = 6x² - 7x + 2. Now, we can equate the coefficients of the corresponding terms on both sides of the equation. This gives us a system of equations: 2a = 6, 2b - a = -7, and -b = 2. Solving this system of equations will give us the values of a and b, and hence the other factor. From the first equation, we get a = 3. From the third equation, we get b = -2. We can verify these values by substituting them into the second equation: 2(-2) - 3 = -7, which is true. Therefore, the other factor is (3x - 2). This step-by-step approach, combining algebraic manipulation and logical deduction, allows us to systematically unveil the other factor of the given quadratic expression.
Verifying the Solution: Ensuring Accuracy and Understanding
Once we have found a potential solution, it is crucial to verify its correctness. This step not only ensures the accuracy of our answer but also deepens our understanding of the factorization process. To verify that (3x - 2) is indeed the other factor of 6x² - 7x + 2, we can simply multiply (2x - 1) and (3x - 2) together and see if we obtain the original quadratic expression. Multiplying (2x - 1) by (3x - 2) using the distributive property (or the FOIL method), we get: (2x - 1)(3x - 2) = 2x(3x) + 2x(-2) - 1(3x) - 1(-2) = 6x² - 4x - 3x + 2 = 6x² - 7x + 2. As we can see, the result of the multiplication is indeed 6x² - 7x + 2, which confirms that (3x - 2) is the correct other factor. This verification process highlights the fundamental relationship between factors and the original polynomial. It reinforces the idea that factorization is essentially the reverse of multiplication. By verifying our solution, we gain confidence in our answer and solidify our understanding of the underlying algebraic principles.
Conclusion: Mastering Factorization and Its Applications
In conclusion, we have successfully navigated the process of finding the other factor of the quadratic expression 6x² - 7x + 2, given that (2x - 1) is one factor. Through a combination of algebraic manipulation, logical deduction, and verification, we have arrived at the solution: (3x - 2). This journey has not only provided us with the answer to a specific problem but has also illuminated the broader concepts of factorization and its importance in algebra. Factorization is a fundamental skill that extends far beyond textbook problems. It is a powerful tool for simplifying expressions, solving equations, and understanding the relationships between polynomials. Mastering factorization opens doors to more advanced topics in mathematics, such as calculus and linear algebra. Moreover, the problem-solving strategies we have employed, such as breaking down complex problems into smaller steps, exploring different methods, and verifying solutions, are valuable skills that can be applied in various fields beyond mathematics. As you continue your mathematical journey, remember that factorization is not just a technique to be memorized but a concept to be understood and applied creatively. With practice and perseverance, you can master factorization and unlock its vast potential.
The correct answer is A. 3x - 2.
