Simplifying Polynomial Expressions A Step By Step Guide To (y² - 3y) × (y² + 10y + 21) / (y² - 9)
Introduction: Delving into Polynomial Expressions
In the realm of algebra, polynomial expressions form the bedrock of many mathematical concepts. Understanding how to manipulate and simplify these expressions is crucial for success in higher-level mathematics and various scientific disciplines. This article aims to dissect and demystify the process of multiplying and simplifying a specific polynomial expression: (y² - 3y) × (y² + 10y + 21) / (y² - 9). We will embark on a step-by-step journey, unraveling the complexities and highlighting the underlying principles that govern polynomial arithmetic. By the end of this exploration, you will not only grasp the solution but also gain a deeper appreciation for the elegance and power of algebraic manipulation. Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. They are fundamental in various fields such as engineering, physics, economics, and computer science, where they are used to model and solve real-world problems. Mastering the manipulation of polynomial expressions, including multiplication and simplification, is essential for anyone seeking a strong foundation in these disciplines. The expression we are about to analyze showcases several important aspects of polynomial operations, such as factoring, multiplication, and division. These operations are the building blocks for solving equations, graphing functions, and tackling more complex mathematical challenges. The ability to confidently navigate these operations will undoubtedly enhance your problem-solving skills and open doors to advanced mathematical concepts.
Step 1: Factoring the Polynomial Expressions
The cornerstone of simplifying polynomial expressions lies in the art of factoring. Factoring is the process of breaking down a polynomial into its constituent factors, which are simpler expressions that, when multiplied together, yield the original polynomial. This process is akin to prime factorization in number theory, where we decompose a number into its prime factors. In our case, we have three polynomial expressions that warrant factoring: y² - 3y, y² + 10y + 21, and y² - 9. Let's tackle each one individually.
Factoring y² - 3y
The expression y² - 3y is a classic example of factoring out a common factor. We observe that both terms, y² and -3y, share a common factor of y. By factoring out y, we can rewrite the expression as y(y - 3). This simple step transforms the quadratic expression into a product of two linear factors. The beauty of factoring lies in its ability to reveal the underlying structure of the polynomial and pave the way for further simplification. Factoring out the greatest common factor is often the first step in simplifying any polynomial expression. It allows us to reduce the complexity of the expression and identify potential cancellations or simplifications in subsequent steps. In this case, factoring out y from y² - 3y makes the expression easier to work with and sets the stage for further manipulation.
Factoring y² + 10y + 21
The expression y² + 10y + 21 is a quadratic trinomial, which can often be factored into the product of two binomials. To factor this expression, we need to find two numbers that add up to 10 (the coefficient of the y term) and multiply to 21 (the constant term). These numbers are 3 and 7. Therefore, we can factor the expression as (y + 3)(y + 7). This factorization is based on the principle that (y + a)(y + b) = y² + (a + b)y + ab. By identifying the appropriate values for a and b, we can effectively factor the quadratic trinomial. Factoring quadratic trinomials is a fundamental skill in algebra, and mastering it opens doors to solving quadratic equations and simplifying more complex expressions. The ability to recognize patterns and apply factoring techniques is crucial for success in algebraic manipulation. In this case, recognizing that 3 and 7 satisfy the required conditions allows us to easily factor the expression.
Factoring y² - 9
The expression y² - 9 is a classic example of the difference of squares. The difference of squares pattern states that a² - b² = (a + b)(a - b). In this case, a = y and b = 3, so we can factor the expression as (y + 3)(y - 3). This factorization is a direct application of a well-known algebraic identity, and it simplifies the expression significantly. Recognizing the difference of squares pattern is a powerful tool in factoring, and it can often lead to quick and efficient solutions. This pattern appears frequently in algebraic problems, and mastering its application is essential for simplifying expressions and solving equations. In this case, recognizing that y² - 9 fits the pattern allows us to factor it effortlessly.
Step 2: Rewriting the Expression with Factored Forms
Now that we have factored each polynomial expression, we can rewrite the original expression in its factored form. This substitution transforms the expression into a more manageable form, where cancellations and simplifications can be readily identified. The original expression was (y² - 3y) × (y² + 10y + 21) / (y² - 9). Substituting the factored forms, we get: y(y - 3) × (y + 3)(y + 7) / (y + 3)(y - 3). This rewritten expression clearly reveals the common factors that can be canceled out, leading to further simplification. Rewriting the expression in its factored form is a crucial step in simplifying rational expressions. It allows us to identify and cancel out common factors, which reduces the complexity of the expression and makes it easier to work with. This process is analogous to simplifying fractions by canceling out common factors in the numerator and denominator.
Step 3: Canceling Common Factors
The beauty of factoring truly shines when we begin to cancel common factors. In the rewritten expression, y(y - 3) × (y + 3)(y + 7) / (y + 3)(y - 3), we observe that the factors (y - 3) and (y + 3) appear in both the numerator and the denominator. These common factors can be canceled out, as any expression divided by itself equals 1. Canceling these factors, we are left with y × (y + 7). This simplification drastically reduces the complexity of the expression, making it easier to understand and manipulate. Canceling common factors is a fundamental technique in simplifying rational expressions. It allows us to reduce the expression to its simplest form by eliminating redundant factors. This process is analogous to reducing a fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor. The ability to identify and cancel common factors is a crucial skill in algebraic manipulation.
Step 4: Simplifying the Expression
After canceling the common factors, we are left with y × (y + 7). To further simplify this expression, we can distribute the y across the terms inside the parentheses. This distribution yields y² + 7y. This final expression is the simplified form of the original polynomial expression. It is a quadratic binomial, which is much simpler and easier to work with than the original expression. Simplifying expressions is a core skill in algebra, and it allows us to reduce complex expressions to their most basic forms. This simplification makes it easier to understand the expression's behavior, solve equations involving the expression, and perform further algebraic manipulations. In this case, distributing the y completes the simplification process and provides a clear and concise representation of the original expression.
Final Result: The Simplified Polynomial
Therefore, the simplified form of the polynomial expression (y² - 3y) × (y² + 10y + 21) / (y² - 9) is y² + 7y. This journey through factoring, canceling, and simplifying highlights the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms. This simplified expression is equivalent to the original expression, but it is easier to understand and work with. The process of simplifying polynomial expressions is not just about finding the correct answer; it's about gaining a deeper understanding of the underlying algebraic principles and developing problem-solving skills that can be applied to a wide range of mathematical challenges. The ability to confidently manipulate polynomial expressions is a valuable asset in mathematics and related fields. By mastering techniques such as factoring, canceling common factors, and distributing terms, you can unlock the power of algebra and tackle more complex problems with ease.
Conclusion: Mastering Polynomial Manipulation
In conclusion, we have successfully navigated the intricacies of multiplying and simplifying the polynomial expression (y² - 3y) × (y² + 10y + 21) / (y² - 9). Through the strategic application of factoring, canceling common factors, and simplifying, we arrived at the elegant solution of y² + 7y. This exercise underscores the importance of mastering fundamental algebraic techniques, which serve as the building blocks for more advanced mathematical concepts. The ability to confidently manipulate polynomial expressions is not merely an academic pursuit; it is a valuable skill that permeates various fields, from engineering and physics to economics and computer science. As you continue your mathematical journey, remember that practice is key. The more you engage with polynomial expressions and apply these techniques, the more proficient you will become. Embrace the challenge, explore the nuances, and unlock the power of algebraic manipulation.
