Unveiling The Quotient Using Synthetic Division: A Step-by-Step Guide

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In the realm of polynomial algebra, synthetic division emerges as a powerful technique for streamlining the process of polynomial division, particularly when the divisor is a linear expression of the form x - a. This method offers a more concise and efficient alternative to long division, allowing us to swiftly determine both the quotient and the remainder. In this article, we embark on a journey to explore the intricacies of synthetic division, demonstrating its application through a concrete example. Our focus will be on unraveling the quotient resulting from the division of the polynomial x⁴ - 10x³ + 32x² - 37x + 10 by the linear expression x - 5. Let's delve into the step-by-step process of synthetic division and unveil the quotient that lies within.

Decoding Synthetic Division: A Step-by-Step Guide

To embark on our quest to decipher the quotient using synthetic division, let's first break down the process into a series of meticulously orchestrated steps:

  1. Setting the Stage: Identifying the Divisor's Root. The cornerstone of synthetic division lies in pinpointing the root of the divisor. In our scenario, the divisor is x - 5, and its root is simply 5. This root will serve as the pivotal value around which our synthetic division calculations will revolve.

  2. Orchestrating Coefficients: Extracting the Polynomial's Essence. Next, we carefully extract the coefficients of the polynomial dividend, ensuring that they are arranged in descending order of their corresponding powers of x. For our dividend, x⁴ - 10x³ + 32x² - 37x + 10, the coefficients gracefully line up as 1, -10, 32, -37, and 10. These coefficients encapsulate the essence of our polynomial, ready to be manipulated by the synthetic division process.

  3. Constructing the Framework: The Synthetic Division Tableau. We now erect the framework for our synthetic division endeavor – a tableau that will guide our calculations. We write the root of the divisor (5) to the left, and the coefficients of the dividend (1, -10, 32, -37, 10) neatly arranged in a row to the right. A horizontal line is drawn beneath the coefficients, creating a designated space for our intermediate calculations and the ultimate quotient.

  4. Initiating the Descent: Bringing Down the Leading Coefficient. The synthetic division process commences with a graceful descent. We bring down the leading coefficient (1) from the dividend's coefficients row, placing it directly below the horizontal line. This marks the beginning of our journey to unveil the quotient.

  5. The Multiplication-Addition Tango: Unveiling the Quotient's Components. The heart of synthetic division lies in an elegant interplay of multiplication and addition. We multiply the root of the divisor (5) by the number we just brought down (1), resulting in 5. This product is then strategically placed beneath the next coefficient in the dividend's row (-10). We then perform addition, summing -10 and 5 to obtain -5. This result is written below the horizontal line, marking our first step in constructing the quotient.

  6. Continuing the Rhythm: Iterating the Process. The multiplication-addition tango continues, as we repeat the previous step for the remaining coefficients. We multiply the root of the divisor (5) by the new number below the line (-5), yielding -25. This product is placed beneath the next coefficient (32), and we add to get 7. We then multiply 5 by 7 to get 35, place it under -37 and add to get -2. Finally, we multiply 5 by -2 to get -10, place it under 10 and add to get 0.

  7. Decoding the Remainder: The Final Verdict. The last number below the horizontal line (0) represents the remainder of the division. In this case, the remainder is 0, signifying that the division is exact – a harmonious conclusion to our synthetic division endeavor.

  8. Unveiling the Quotient: Constructing the Polynomial. The remaining numbers below the horizontal line, excluding the remainder, represent the coefficients of the quotient polynomial. In our case, these numbers are 1, -5, 7, and -2. Since the original dividend was a fourth-degree polynomial and we divided by a first-degree polynomial, the quotient will be a third-degree polynomial. Therefore, the quotient polynomial is x³ - 5x² + 7x - 2.

The Quotient Unveiled: A Concrete Example

Now, let's apply the synthetic division steps to our specific example: dividing x⁴ - 10x³ + 32x² - 37x + 10 by x - 5.

  1. Root of the Divisor: The root of x - 5 is 5.
  2. Coefficients of the Dividend: The coefficients of x⁴ - 10x³ + 32x² - 37x + 10 are 1, -10, 32, -37, and 10.
  3. Synthetic Division Tableau:
5 | 1 -10 32 -37 10
  |________________________
  1. Bringing Down the Leading Coefficient:
5 | 1 -10 32 -37 10
  |________________________
  1
  1. Multiplication-Addition Tango:
5 | 1 -10 32 -37 10
  |     5 -25  35 -10
  |________________________
  1 -5   7  -2  0
  1. Remainder and Quotient: The remainder is 0, and the coefficients of the quotient are 1, -5, 7, and -2.

Therefore, the quotient of x⁴ - 10x³ + 32x² - 37x + 10 divided by x - 5 is x³ - 5x² + 7x - 2.

Conclusion: Mastering the Art of Synthetic Division

Synthetic division stands as a testament to the elegance and efficiency that can be achieved in mathematical operations. By mastering this technique, we gain a powerful tool for simplifying polynomial division, particularly when dealing with linear divisors. The step-by-step process, involving the identification of the divisor's root, the extraction of dividend coefficients, the construction of the tableau, and the rhythmic multiplication-addition tango, culminates in the unveiling of the quotient and the remainder. In our exploration, we successfully determined that the quotient of x⁴ - 10x³ + 32x² - 37x + 10 divided by x - 5 is x³ - 5x² + 7x - 2. This mastery of synthetic division empowers us to tackle more complex polynomial manipulations with confidence and precision.

Therefore, the correct answer is A. x³ - 5x² + 7x - 2.