Finding The Value Of P When A Polynomial Is Divided - A Detailed Solution
Introduction
In the realm of mathematics, polynomial division stands as a fundamental concept with wide-ranging applications. This exploration delves into a specific scenario involving polynomial division and remainder theorem to find the unknown value of 'p'. We are given the expression (x+1)(x-1)(4x+p), which is divided by 2x-1, and the remainder is -3. Our objective is to determine the exact value of 'p' from the given options. To accomplish this, we will meticulously apply the remainder theorem and demonstrate the step-by-step solution, ensuring clarity and comprehension for all learners. Polynomial division is the cornerstone of algebraic manipulations, and mastering it will enable problem-solving in various mathematical contexts. In this article, we will explain a step-by-step solution to determine the value of 'p'.
Understanding the Remainder Theorem
The remainder theorem is a powerful tool that provides a shortcut for finding the remainder when a polynomial is divided by a linear factor. This theorem states that if a polynomial f(x) is divided by (x - c), then the remainder is f(c). In simpler terms, to find the remainder, we substitute the value that makes the divisor zero into the polynomial. This theorem avoids the lengthy process of polynomial long division, making problem-solving more efficient. The remainder theorem is not just a computational tool but a conceptual bridge connecting polynomial division and function evaluation. Its applications extend beyond basic algebra, finding relevance in calculus and other higher mathematics areas. It enables mathematicians to predict the outcome of division without performing the actual division, a particularly useful skill in complex calculations and theoretical proofs. Understanding the theoretical underpinnings of the remainder theorem is crucial for mastering polynomial algebra and its applications in various mathematical contexts. In this particular problem, we will apply the remainder theorem to find the value of 'p', making it a cornerstone of our solution.
Detailed Solution Steps
Let's embark on a detailed step-by-step solution to determine the value of 'p' when (x+1)(x-1)(4x+p) is divided by 2x-1, leaving a remainder of -3. First, we denote the polynomial as f(x) = (x+1)(x-1)(4x+p). The divisor is 2x - 1. To apply the remainder theorem, we need to find the value of x that makes the divisor equal to zero. Solving 2x - 1 = 0, we get x = 1/2. Next, we substitute x = 1/2 into the polynomial f(x):
f(1/2) = (1/2 + 1)(1/2 - 1)(4(1/2) + p)
Simplifying the expression, we have:
f(1/2) = (3/2)(-1/2)(2 + p)
According to the remainder theorem, f(1/2) is the remainder when f(x) is divided by 2x - 1. We are given that the remainder is -3. Therefore, we set f(1/2) = -3:
(3/2)(-1/2)(2 + p) = -3
This simplifies to:
(-3/4)(2 + p) = -3
To solve for p, we multiply both sides by -4/3:
2 + p = (-3)(-4/3)
2 + p = 4
Subtracting 2 from both sides, we get:
p = 4 - 2
p = 2
However, upon closer examination of the question and potential answer options (A. 4, B. -4, C. 5, D. -5), there seems to be a mistake in the question itself or in my calculation. Let's backtrack and check the calculation again to make sure every step is correct.
f(1/2) = (3/2)(-1/2)(2 + p) = -3
(-3/4)(2 + p) = -3
Multiply both sides by -4/3:
2 + p = 4
Subtract 2 from both sides:
p = 2
It appears there was no mistake in the calculation. The value of p indeed equals 2, which doesn't match any of the given options. This suggests that either the options are incorrect or there might be an error in the problem statement itself. If we proceed with the closest integer value from the options, considering potential rounding errors or typographical issues, we might infer a potential mistake in the options provided. Given the discrepancy, it is important to highlight the exact calculation steps to ensure transparency and accuracy in our approach. If the options provided are accurate representations, then the problem statement or the intended solution might need further review. In this scenario, the most appropriate step is to acknowledge the discrepancy and provide the correct calculated value alongside the outlined steps, allowing for potential reevaluation of the problem statement or answer choices.
Conclusion
In conclusion, when we divide (x+1)(x-1)(4x+p) by 2x-1 and the remainder is -3, our meticulous step-by-step solution using the remainder theorem reveals that p = 2. This value, however, does not align with the provided options (A. 4, B. -4, C. 5, D. -5), indicating a potential discrepancy either in the question's options or the question itself. The remainder theorem has proven to be an efficient method for determining the value of 'p' without resorting to polynomial long division. This exploration has not only reinforced our understanding of polynomial division but also highlighted the importance of verifying solutions against given options to identify potential inconsistencies. The ability to identify such discrepancies is a crucial skill in mathematical problem-solving, ensuring accuracy and promoting critical thinking. Despite the mismatch, the process of arriving at the solution has solidified our grasp on the underlying mathematical principles and their applications. This methodical approach is essential in mathematical problem-solving, and we trust it will aid in addressing similar challenges with confidence and precision. We encourage readers to practice such problems, emphasizing the significance of meticulous calculation and cross-validation to avoid potential pitfalls. The correct application of theorems and methods remains paramount in the quest for accurate mathematical solutions.
