Multiplying And Simplifying Rational Expressions

In mathematics, particularly in algebra, simplifying expressions is a fundamental skill. This article delves into the process of multiplying rational expressions and simplifying the result. We will tackle the specific problem of multiplying (x-3)/(4x-4) by (x^2-3x+2)/(x+1), providing a step-by-step solution and highlighting key concepts. Understanding these concepts is crucial for anyone studying algebra or calculus, as they form the basis for more complex mathematical operations. Mastering the manipulation of rational expressions not only enhances your problem-solving skills but also provides a solid foundation for advanced mathematical concepts.

Understanding Rational Expressions

Before we dive into the multiplication and simplification process, let's define what rational expressions are. A rational expression is simply a fraction where the numerator and the denominator are polynomials. Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include x^2 + 3x - 2, 4x - 4, and x + 1. Therefore, a rational expression can look like (x^2 + 3x - 2) / (4x - 4) or any similar fraction with polynomials in the numerator and denominator. Rational expressions are ubiquitous in algebra and calculus. They appear in various contexts, such as solving equations, graphing functions, and modeling real-world phenomena. Understanding how to manipulate these expressions is paramount for success in higher-level mathematics.

The importance of understanding rational expressions extends beyond the classroom. In fields like engineering, physics, and economics, rational expressions are used to model various relationships and solve practical problems. For instance, in electrical engineering, they can represent the impedance of a circuit; in physics, they might describe the motion of an object; and in economics, they can model supply and demand curves. The ability to simplify and work with rational expressions allows professionals to analyze and solve complex problems in these fields. Therefore, a solid grasp of rational expressions is not just beneficial for academic pursuits but also for various professional careers.

Multiplying Rational Expressions: A Step-by-Step Guide

To multiply rational expressions, the basic principle is similar to multiplying regular fractions: multiply the numerators together and multiply the denominators together. However, with rational expressions, it's often beneficial to factor the polynomials first. Factoring allows us to identify common factors between the numerator and the denominator, which can then be canceled out to simplify the expression. This simplification process is crucial for obtaining the most concise form of the answer. Let's apply this to our specific problem: (x-3)/(4x-4) * (x^2-3x+2)/(x+1).

The first step is to factor the polynomials in both the numerators and the denominators. In the first fraction, the numerator (x-3) is already in its simplest form and cannot be factored further. The denominator (4x-4) can be factored by taking out the common factor of 4, resulting in 4(x-1). In the second fraction, the numerator (x^2-3x+2) is a quadratic expression that can be factored into (x-1)(x-2). The denominator (x+1) is already in its simplest form and cannot be factored further. Now, we rewrite the expression with the factored polynomials: [(x-3) / 4(x-1)] * [(x-1)(x-2) / (x+1)]. This factored form makes it easier to identify common factors that can be canceled out in the next step.

Simplifying the Result

After factoring the polynomials, the next crucial step is simplification. Simplification involves canceling out any common factors that appear in both the numerator and the denominator. This process is based on the principle that dividing both the numerator and denominator of a fraction by the same non-zero value does not change the value of the fraction. In our expression, [(x-3) / 4(x-1)] * [(x-1)(x-2) / (x+1)], we can see that the factor (x-1) appears in both the numerator and the denominator. This common factor can be canceled out, which simplifies the expression significantly. Canceling common factors is a fundamental technique in simplifying rational expressions, and it's essential to perform this step before presenting the final answer.

After canceling the common factor (x-1), our expression becomes [(x-3) / 4] * [(x-2) / (x+1)]. Now, we multiply the remaining numerators together and the remaining denominators together. This gives us (x-3)(x-2) in the numerator and 4(x+1) in the denominator. The resulting expression is [(x-3)(x-2)] / [4(x+1)]. To further simplify, we can expand the numerator by multiplying the two binomials (x-3) and (x-2). This expansion gives us x^2 - 2x - 3x + 6, which simplifies to x^2 - 5x + 6. Therefore, the simplified expression is (x^2 - 5x + 6) / [4(x+1)]. This is the most simplified form of the original expression, where all possible cancellations and multiplications have been performed.

Final Answer and Conclusion

After multiplying the rational expressions and simplifying the result, we arrive at the final answer: (x^2 - 5x + 6) / [4(x+1)]. This represents the most simplified form of the expression obtained by multiplying (x-3)/(4x-4) by (x^2-3x+2)/(x+1). The process involved factoring the polynomials, canceling common factors, and then multiplying the remaining terms. This step-by-step approach is essential for accurately simplifying rational expressions. Mastering these techniques is not only crucial for academic success in mathematics but also for various practical applications in science, engineering, and other fields.

In conclusion, multiplying and simplifying rational expressions is a fundamental skill in algebra. By understanding the principles of factoring, canceling common factors, and multiplying polynomials, one can effectively simplify complex expressions. The ability to manipulate rational expressions is a cornerstone of algebraic proficiency and serves as a building block for more advanced mathematical concepts. The problem we tackled in this article provides a clear example of how to approach such problems systematically. By following the steps outlined, you can confidently multiply and simplify rational expressions, enhancing your mathematical problem-solving skills and preparing you for more challenging topics in the future.

1. Factor the polynomials: Identify and factor any polynomials in the numerators and denominators.
2. Cancel common factors: Look for factors that appear in both the numerator and the denominator and cancel them out.
3. Multiply: Multiply the remaining numerators together and the remaining denominators together.
4. Simplify: Expand and simplify the resulting expression if necessary.

By following these steps, you can confidently multiply and simplify rational expressions in various mathematical contexts.