Remainder Theorem Find Quotient With Remainder 20 Math Problem

In this comprehensive guide, we will delve into the fascinating world of the Remainder Theorem and explore its applications in polynomial division. The Remainder Theorem provides a powerful shortcut for determining the remainder when a polynomial is divided by a linear expression, without actually performing the long division. This theorem is not only a fundamental concept in algebra but also a valuable tool for solving various problems involving polynomials. Our primary focus will be on applying the Remainder Theorem to identify quotients that yield a specific remainder, such as 20 in the given problem. We will break down the theorem, illustrate its use with examples, and meticulously analyze the provided options to arrive at the correct answer. This journey will not only enhance your understanding of the Remainder Theorem but also sharpen your problem-solving skills in polynomial algebra.

Demystifying the Remainder Theorem

The Remainder Theorem is a cornerstone of polynomial algebra, offering a direct method to find the remainder of a polynomial division without resorting to long division. The theorem states that if a polynomial f(x) is divided by a linear divisor (x - c), the remainder is f(c). This seemingly simple concept is incredibly powerful. To illustrate, consider dividing a polynomial like f(x) = x² + 3x + 2 by (x - 1). Instead of performing the division, we can evaluate f(1), which gives us 1² + 3(1) + 2 = 6. Thus, the remainder is 6. This principle holds true for polynomials of any degree and makes the Remainder Theorem an indispensable tool in various algebraic manipulations.

The elegance of the Remainder Theorem lies in its efficiency. It transforms a division problem into a simple evaluation. The theorem is not just a computational shortcut; it provides deep insights into the relationship between polynomial roots and remainders. For instance, if f(c) = 0, it implies that (x - c) is a factor of f(x), a concept central to the Factor Theorem, which is a direct extension of the Remainder Theorem. Furthermore, understanding the Remainder Theorem is crucial for grasping more advanced topics like synthetic division and polynomial factorization. In essence, the Remainder Theorem is more than a formula; it's a bridge connecting various concepts in polynomial algebra, making complex problems more accessible and understandable.

Applying the Remainder Theorem: A Practical Approach

To effectively utilize the Remainder Theorem, it's essential to understand its practical application. The process involves two key steps: identifying the value of c from the divisor (x - c) and evaluating the polynomial f(x) at x = c. Let's consider a detailed example to solidify this understanding. Suppose we want to find the remainder when f(x) = 2x³ - 5x² + 3x - 10 is divided by (x - 3). Here, c = 3. We then evaluate f(3), which is 2(3)³ - 5(3)² + 3(3) - 10. Calculating this gives us 2(27) - 5(9) + 9 - 10 = 54 - 45 + 9 - 10 = 8. Therefore, the remainder is 8. This process is significantly less cumbersome than performing long division, especially for higher-degree polynomials.

Another crucial aspect is recognizing the sign convention in the divisor. If the divisor is in the form (x + c), it is equivalent to (x - (-c)), meaning we need to evaluate the polynomial at x = -c. For example, if the divisor is (x + 2), then c = -2. This subtle but important detail is often a source of errors. The Remainder Theorem is not limited to numerical remainders; it can also provide insights into the nature of polynomial roots and factors. If the remainder is zero, it indicates that the divisor is a factor of the polynomial, a principle that forms the basis of the Factor Theorem. Understanding these nuances of the Remainder Theorem empowers you to tackle a wide range of polynomial problems efficiently and accurately, making it an indispensable tool in your mathematical arsenal.

Analyzing the Options Using the Remainder Theorem

Now, let's apply our understanding of the Remainder Theorem to the given options and identify the quotient that yields a remainder of 20. We will systematically evaluate each option, focusing on the core steps of the theorem: identifying the value of c from the divisor and evaluating the polynomial at x = c. This process will not only lead us to the correct answer but also reinforce the practical application of the theorem.

Option A: $\left(3 x^3+7 x^2+5 x+2\right) \div(x-2)$

In this option, the polynomial is f(x) = 3x³ + 7x² + 5x + 2, and the divisor is (x - 2). Thus, c = 2. We now need to evaluate f(2). This gives us 3(2)³ + 7(2)² + 5(2) + 2. Calculating this expression, we get 3(8) + 7(4) + 10 + 2 = 24 + 28 + 10 + 2 = 64. Therefore, the remainder for option A is 64, which is not 20. This eliminates option A as the correct answer. This methodical approach is crucial in applying the Remainder Theorem effectively. We are not just looking for any remainder; we are specifically seeking a remainder of 20, which requires precise calculations.

Option B: $\left(x^3+5 x^2-4 x+6\right) \div(x+5)$

For option B, the polynomial is f(x) = x³ + 5x² - 4x + 6, and the divisor is (x + 5). Here, c = -5 (remember to consider the sign). We evaluate f(-5), which is (-5)³ + 5(-5)² - 4(-5) + 6. This simplifies to -125 + 5(25) + 20 + 6 = -125 + 125 + 20 + 6 = 26. The remainder for option B is 26, which is also not 20. Consequently, option B is not the correct answer. The process of evaluating polynomials at negative values highlights the importance of careful calculation and attention to detail in applying the Remainder Theorem.

Option C: $\left(3 x^4-5 x^3+5 x+2\right) \div(x-2)$

In option C, we have the polynomial f(x) = 3x⁴ - 5x³ + 5x + 2, and the divisor is (x - 2). Therefore, c = 2. Evaluating f(2), we get 3(2)⁴ - 5(2)³ + 5(2) + 2. This calculates to 3(16) - 5(8) + 10 + 2 = 48 - 40 + 10 + 2 = 20. Aha! The remainder for option C is 20, which matches our target remainder. This indicates that option C is likely the correct answer. However, to be absolutely certain, we should still examine option D to rule out any possibility of multiple correct answers.

Option D: $\left(x^4-8 x^2+16\right) \div(x-2)$

For option D, the polynomial is f(x) = x⁴ - 8x² + 16, and the divisor is (x - 2). Thus, c = 2. Evaluating f(2) gives us (2)⁴ - 8(2)² + 16. This simplifies to 16 - 8(4) + 16 = 16 - 32 + 16 = 0. The remainder for option D is 0, which is not 20. Therefore, option D is not the correct answer. This final evaluation confirms that option C is indeed the only option with a remainder of 20.

Conclusion: The Correct Quotient

After a thorough analysis of all the options using the Remainder Theorem, we have confidently identified the correct answer. By systematically evaluating each polynomial at the value derived from its divisor, we were able to determine the remainder for each option. This methodical approach is a testament to the power and efficiency of the Remainder Theorem in solving polynomial division problems.

The Verdict: Option C is the Answer

Our step-by-step evaluation has definitively shown that option C, $\left(3 x^4-5 x^3+5 x+2\right) \div(x-2)$, yields a remainder of 20 when the Remainder Theorem is applied. This conclusion is not just a result of calculation; it's a demonstration of the theorem's practical utility. The Remainder Theorem allows us to bypass the complexities of long division and directly find the remainder, saving time and reducing the chance of errors.

In summary, the Remainder Theorem is a powerful tool in polynomial algebra. It provides a straightforward method for finding the remainder of a polynomial division, and its applications extend beyond simple calculations. Understanding and mastering this theorem is crucial for anyone delving into the world of algebra and polynomial functions. By applying the theorem methodically and paying attention to detail, as we did in this analysis, you can confidently tackle a wide range of polynomial problems and unlock the beauty and elegance of mathematics.