Deciphering The Quotient Of Rational Expressions A Step By Step Guide
Navigating the realm of algebraic expressions can often feel like traversing a labyrinth, especially when encountering complex fractions and division operations. In this comprehensive exploration, we will embark on a step-by-step journey to decipher the quotient of the following expression:
\frac{2y^2 - 6y - 20}{4y + 12} \div \frac{y^2 + 5y + 6}{3y^2 + 18y + 27}
This intricate expression, a tapestry woven with polynomials and division, demands a meticulous approach to unravel its hidden quotient. We will meticulously dissect each component, employing a symphony of algebraic techniques, to arrive at the simplified result. Our expedition will involve factoring, strategic cancellation, and a deep understanding of the fundamental principles governing rational expressions.
Initial Transformation: Division to Multiplication
At the heart of our algebraic odyssey lies the division operation. The initial step in conquering this expression is to transform the division into its multiplicative counterpart. We achieve this by inverting the second fraction, the divisor, and replacing the division symbol with multiplication. This fundamental transformation sets the stage for simplification.
Our expression now takes on a new form:
\frac{2y^2 - 6y - 20}{4y + 12} \times \frac{3y^2 + 18y + 27}{y^2 + 5y + 6}
By transforming division into multiplication, we have paved the way for a more streamlined approach to simplification. The expression now presents itself as a product of two rational expressions, inviting us to explore the possibilities of factorization and cancellation.
Factorization: Unveiling the Building Blocks
The cornerstone of simplifying rational expressions lies in the art of factorization. We must meticulously factor each polynomial, dissecting them into their fundamental building blocks. This process allows us to identify common factors that can be strategically canceled, leading us closer to the simplified quotient.
Factoring the Numerator of the First Fraction
Our attention now turns to the numerator of the first fraction, the quadratic expression 2y² - 6y - 20. We embark on a quest to factor out the greatest common factor (GCF), revealing a hidden simplicity within. The GCF in this case is 2, which we gracefully extract:
2(y^2 - 3y - 10)
With the GCF extracted, we delve deeper into the quadratic expression y² - 3y - 10. Our mission is to express this quadratic as a product of two binomials. We seek two numbers that dance harmoniously, multiplying to -10 and summing to -3. After careful consideration, we identify the pair -5 and 2. Thus, we factor the quadratic as:
2(y - 5)(y + 2)
The numerator of the first fraction now stands revealed in its factored form, a testament to our algebraic prowess.
Factoring the Denominator of the First Fraction
Our focus shifts to the denominator of the first fraction, the linear expression 4y + 12. Here, too, we seek the GCF, a beacon of simplification. In this case, the GCF is 4, which we deftly factor out:
4(y + 3)
The denominator of the first fraction now stands elegantly factored, a testament to the power of simplification.
Factoring the Numerator of the Second Fraction
We now turn our attention to the numerator of the second fraction, the quadratic expression 3y² + 18y + 27. Once again, we embark on a quest to extract the GCF, seeking a pathway to simplification. The GCF in this instance is 3, which we gracefully factor out:
3(y^2 + 6y + 9)
With the GCF extracted, we delve into the quadratic expression y² + 6y + 9. We recognize this as a perfect square trinomial, a harmonious expression that can be factored as:
3(y + 3)^2
The numerator of the second fraction now stands proudly factored, a testament to our keen algebraic eye.
Factoring the Denominator of the Second Fraction
Our gaze now settles upon the denominator of the second fraction, the quadratic expression y² + 5y + 6. We seek to express this quadratic as a product of two binomials. We seek two numbers that, when multiplied, yield 6, and when summed, yield 5. After careful deliberation, we identify the pair 2 and 3. Thus, we factor the quadratic as:
(y + 2)(y + 3)
The denominator of the second fraction now stands beautifully factored, completing our factorization endeavor.
The Expression in its Factored Glory
With all the components factored, our expression now stands revealed in its full factored splendor:
\frac{2(y - 5)(y + 2)}{4(y + 3)} \times \frac{3(y + 3)^2}{(y + 2)(y + 3)}
Each polynomial has been meticulously dissected, its building blocks exposed. The stage is now set for strategic cancellation, where common factors will gracefully vanish, leading us closer to the simplified quotient.
Cancellation: A Symphony of Simplification
The heart of simplifying rational expressions lies in the art of cancellation. We seek common factors that grace both the numerator and denominator, poised to be elegantly eliminated. This strategic cancellation unveils the underlying simplicity of the expression.
Identifying Common Factors
Our eyes dance across the expression, identifying the common factors that beckon cancellation. We observe the factor (y + 2) appearing in both the numerator and denominator, ready to vanish in a graceful simplification. Similarly, we spot the factor (y + 3) gracing both the numerator and denominator, poised for elimination.
The Dance of Cancellation
With a flourish, we cancel the common factors, watching them disappear in a symphony of simplification:
- The factor (y + 2) in the numerator of the first fraction cancels with the factor (y + 2) in the denominator of the second fraction.
- One instance of the factor (y + 3) in the denominator of the first fraction cancels with one instance of the factor (y + 3) in the numerator of the second fraction.
The Simplified Expression
With the cancellation complete, our expression now stands transformed, its complexity significantly reduced:
\frac{2(y - 5)}{4} \times \frac{3(y + 3)}{1}
Further simplification beckons, as we observe the numerical factors that can be reduced.
Further Simplification: Numerical Refinement
Our quest for simplification continues as we turn our attention to the numerical factors. We observe that the fraction 2/4 can be elegantly simplified to 1/2. This numerical refinement brings us closer to the final quotient.
The Simplified Fraction
With the numerical simplification performed, our expression takes on a more polished form:
\frac{(y - 5)}{2} \times 3(y + 3)
We now stand poised to combine the remaining factors, unveiling the final quotient.
The Final Quotient: A Triumph of Simplification
Our algebraic journey culminates in the revelation of the final quotient. We combine the remaining factors, multiplying the numerators and denominators:
\frac{3(y - 5)(y + 3)}{2}
This elegant expression represents the simplified quotient of the original complex rational expression. Our meticulous factorization, strategic cancellation, and numerical refinement have led us to this triumphant conclusion.
The Answer
Comparing our simplified quotient with the given options, we find a subtle difference. Our expression, \frac{3(y - 5)(y + 3)}{2}, does not directly match any of the provided choices. However, we recognize that option B, \frac{3(y-5)}{2}, is the closest match. A careful review of our steps reveals a minor oversight: we failed to cancel the (y+3) term in the numerator with one of the (y+3) terms in the denominator during the cancellation phase.
Correcting this oversight, we arrive at the final, fully simplified quotient:
\frac{3(y - 5)}{2}
Therefore, the correct answer is:
B.
This journey through algebraic simplification has been a testament to the power of meticulous technique and unwavering attention to detail. We have successfully navigated the labyrinth of rational expressions, emerging with the simplified quotient as our prize.
Conclusion
In this comprehensive exploration, we have meticulously dissected a complex rational expression, unraveling its quotient through a symphony of algebraic techniques. We have witnessed the power of factorization, the elegance of cancellation, and the refinement of numerical simplification. This journey serves as a testament to the beauty and precision of mathematics, where intricate expressions can be transformed into elegant simplicity. Understanding the quotient of rational expressions is not just an exercise in algebra; it's a journey into the heart of mathematical problem-solving, enhancing analytical skills and fostering a deeper appreciation for the interconnectedness of mathematical concepts. Mastering the process of simplifying rational expressions opens doors to more advanced mathematical concepts, laying a solid foundation for future explorations in calculus, differential equations, and beyond. This skill is not just for academics; it's a valuable asset in various fields, from engineering to economics, where complex relationships often need to be distilled into simpler, manageable forms. The ability to factor polynomials and cancel common factors is crucial in these scenarios, allowing for more efficient problem-solving and decision-making. Remember, practice is key to mastering these techniques. The more you engage with rational expressions, the more comfortable you'll become with identifying patterns, applying the right strategies, and arriving at the correct solution. Each problem is an opportunity to refine your skills and deepen your understanding. So, embrace the challenge, explore the intricacies, and let the beauty of mathematics guide you to elegant solutions.