Polynomial Division How To Find The Quotient Of (3x³ + 4x - 32) ÷ (x - 2)

by ADMIN 74 views
Iklan Headers

Introduction

In mathematics, polynomial division is a fundamental operation, crucial for simplifying expressions, solving equations, and understanding the behavior of polynomial functions. This article delves into the process of dividing polynomials, specifically focusing on finding the quotient of the expression (3x³ + 4x - 32) ÷ (x - 2). We will explore the steps involved in polynomial long division and provide a detailed, step-by-step solution to this problem. Understanding polynomial division is essential for students and professionals in various fields, including algebra, calculus, and engineering. By mastering this technique, you can enhance your problem-solving skills and gain a deeper appreciation for mathematical concepts.

Understanding Polynomial Division

Polynomial division, at its core, is similar to the long division you learned in arithmetic. However, instead of dividing numbers, we are dividing algebraic expressions that involve variables and exponents. The key idea is to break down the dividend (the polynomial being divided) into smaller parts that can be evenly divided by the divisor (the polynomial doing the dividing). This process yields a quotient (the result of the division) and a remainder (any part of the dividend that cannot be evenly divided). Polynomial long division is a systematic method for achieving this, providing a clear and organized way to perform the division. Mastering this technique is not just about finding the right answer; it’s about understanding the underlying principles of algebra and how polynomials interact. With practice, polynomial division becomes a powerful tool in your mathematical arsenal, enabling you to tackle more complex problems with confidence.

Setting Up the Polynomial Long Division

To begin the process of polynomial long division, we first need to set up the problem correctly. This involves writing the dividend (3x³ + 4x - 32) and the divisor (x - 2) in the appropriate format. Pay close attention to the placeholders for missing terms. In the dividend, notice that there is no x² term. To maintain the correct structure for long division, we need to include a placeholder term, 0x², in the dividend. This ensures that we align like terms correctly during the division process. The setup will look like this:

 x - 2 | 3x³ + 0x² + 4x - 32

Setting up the problem correctly is crucial because it helps prevent errors and keeps the division process organized. This initial step sets the stage for the entire solution, making the subsequent calculations more manageable. Always double-check the setup before proceeding to the next step to ensure accuracy and avoid unnecessary mistakes.

Step-by-Step Solution to (3x³ + 4x - 32) ÷ (x - 2)

Now, let's walk through the step-by-step solution to find the quotient of (3x³ + 4x - 32) ÷ (x - 2) using polynomial long division. This process will illustrate how to methodically divide polynomials, ensuring a clear understanding of each step.

Step 1: Divide the first term of the dividend by the first term of the divisor.

  • We start by dividing 3x³ (the first term of the dividend) by x (the first term of the divisor). This gives us 3x².

Step 2: Multiply the quotient term by the entire divisor.

  • Multiply 3x² by the entire divisor (x - 2): 3x² * (x - 2) = 3x³ - 6x².

Step 3: Subtract the result from the dividend.

  • Subtract (3x³ - 6x²) from (3x³ + 0x²): (3x³ + 0x²) - (3x³ - 6x²) = 6x².

Step 4: Bring down the next term from the dividend.

  • Bring down the next term, +4x, from the original dividend. The new expression becomes 6x² + 4x.

Step 5: Repeat the process.

  • Divide 6x² by x, which gives us +6x.
  • Multiply 6x by (x - 2): 6x * (x - 2) = 6x² - 12x.
  • Subtract (6x² - 12x) from (6x² + 4x): (6x² + 4x) - (6x² - 12x) = 16x.

Step 6: Bring down the last term from the dividend.

  • Bring down the last term, -32. The expression becomes 16x - 32.

Step 7: Repeat the process one final time.

  • Divide 16x by x, which gives us +16.
  • Multiply 16 by (x - 2): 16 * (x - 2) = 16x - 32.
  • Subtract (16x - 32) from (16x - 32): (16x - 32) - (16x - 32) = 0.

Step 8: Determine the quotient and remainder.

  • The quotient is the sum of the terms we obtained in the division process: 3x² + 6x + 16.
  • The remainder is 0, indicating that (x - 2) divides evenly into (3x³ + 4x - 32).

Thus, the quotient of (3x³ + 4x - 32) ÷ (x - 2) is 3x² + 6x + 16.

Detailed Breakdown of the Long Division Process

To further clarify the long division process, let's break down each step with a detailed explanation. This will help solidify your understanding and provide a clear picture of how each term is derived.

  1. Initial Setup: As mentioned earlier, we begin by setting up the problem with the dividend (3x³ + 0x² + 4x - 32) inside the division symbol and the divisor (x - 2) outside. The 0x² term is crucial for maintaining alignment of like terms.
  2. First Division:
    • We divide the first term of the dividend (3x³) by the first term of the divisor (x). This gives us 3x². We write this term above the division symbol, aligned with the x² term in the dividend.
    • Next, we multiply this quotient term (3x²) by the entire divisor (x - 2), resulting in 3x³ - 6x².
    • We subtract this result from the corresponding terms in the dividend: (3x³ + 0x²) - (3x³ - 6x²) = 6x². This subtraction step is vital for reducing the dividend and moving towards the solution.
  3. Bringing Down the Next Term:
    • We bring down the next term from the dividend, which is +4x. This creates a new expression: 6x² + 4x.
  4. Second Division:
    • Now, we divide the first term of the new expression (6x²) by the first term of the divisor (x), which gives us +6x. We add this term to the quotient above the division symbol.
    • We multiply this quotient term (6x) by the entire divisor (x - 2), resulting in 6x² - 12x.
    • We subtract this result from the current expression: (6x² + 4x) - (6x² - 12x) = 16x. Again, subtraction is key to simplifying the expression.
  5. Bringing Down the Last Term:
    • We bring down the final term from the dividend, which is -32. This gives us the expression 16x - 32.
  6. Final Division:
    • We divide the first term of this expression (16x) by the first term of the divisor (x), which gives us +16. We add this term to the quotient above the division symbol.
    • We multiply this quotient term (16) by the entire divisor (x - 2), resulting in 16x - 32.
    • We subtract this result from the current expression: (16x - 32) - (16x - 32) = 0. The remainder is 0, indicating a clean division.
  7. Determining the Quotient:
    • The quotient is the sum of the terms we wrote above the division symbol: 3x² + 6x + 16.

By meticulously following these steps, you can successfully perform polynomial long division and find the quotient of any two polynomials. This detailed breakdown serves as a guide to understanding each part of the process, ensuring accuracy and clarity.

Why Polynomial Division Matters

Understanding polynomial division is not just an academic exercise; it has significant practical applications in various fields. Polynomials are used to model a wide range of real-world phenomena, from the trajectory of a projectile to the behavior of electrical circuits. Polynomial division allows us to simplify these models, making them easier to analyze and understand.

For instance, in engineering, polynomial division can be used to determine the stability of a system or to design control systems. In computer graphics, polynomials are used to create smooth curves and surfaces, and polynomial division can help optimize these curves for performance. In economics, polynomial functions can model supply and demand curves, and division can help in analyzing market equilibrium.

Moreover, polynomial division is a crucial tool in advanced mathematical studies, such as calculus and abstract algebra. It forms the basis for understanding concepts like factoring, roots of polynomials, and rational functions. By mastering polynomial division, you lay a solid foundation for further mathematical exploration and problem-solving.

In summary, the ability to perform polynomial division is a valuable skill that extends beyond the classroom. It enhances your analytical abilities, provides a deeper understanding of mathematical concepts, and opens doors to various practical applications in science, engineering, and beyond.

Conclusion

In conclusion, finding the quotient of (3x³ + 4x - 32) ÷ (x - 2) using polynomial division is a fundamental skill in mathematics. This article has provided a comprehensive guide, breaking down the process into manageable steps. We've seen how to set up the problem, perform the long division, and interpret the results. The quotient, in this case, is 3x² + 6x + 16, with no remainder. This detailed exploration not only answers the specific question but also equips you with the knowledge to tackle similar problems confidently.

Polynomial division is more than just a mechanical process; it's a window into the structure of polynomials and their behavior. By understanding this technique, you gain a deeper appreciation for algebraic concepts and enhance your problem-solving abilities. Remember to practice regularly to solidify your understanding and build proficiency. With a solid grasp of polynomial division, you'll be well-prepared to tackle more advanced mathematical challenges.

Answer

The correct answer is C. 3x² + 6x + 16.