Polynomial Division Explained Solving (2x² + 7x - 15) ÷ (x + 5)

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Introduction: Polynomial Division and Its Significance

In the realm of mathematics, polynomial division stands as a fundamental operation, particularly vital in algebra and calculus. It mirrors the long division method we use with numbers, but instead of digits, we're working with expressions containing variables and exponents. Mastering polynomial division unlocks the door to simplifying complex expressions, solving equations, and understanding the behavior of polynomial functions. This article delves into the solution of the polynomial division problem (2x² + 7x - 15) ÷ (x + 5), employing both long division and synthetic division methods to provide a comprehensive understanding. Polynomial division is a cornerstone of algebraic manipulation, allowing us to break down complex expressions into simpler, more manageable forms. This is crucial for various applications, including finding the roots of polynomials, simplifying rational expressions, and even in calculus when dealing with integration and differentiation. The ability to confidently perform polynomial division is a significant asset in any mathematical endeavor.

The Division Problem: (2x² + 7x - 15) ÷ (x + 5)

The specific problem we aim to solve is the division of the quadratic polynomial 2x² + 7x - 15 by the linear binomial x + 5. This type of problem frequently appears in algebra courses and standardized tests, making it essential to grasp the underlying concepts and techniques. The dividend, 2x² + 7x - 15, is the polynomial being divided, while the divisor, x + 5, is the expression we are dividing by. The quotient, which we aim to find, is the result of this division, and there might also be a remainder, representing any portion of the dividend that is not perfectly divisible by the divisor. Understanding the components of polynomial division, namely the dividend, divisor, quotient, and remainder, is paramount to tackling these problems effectively. Before diving into the solution methods, it's crucial to recognize that polynomial division is not merely a mechanical process; it's a systematic way of unraveling the relationship between different polynomial expressions. By finding the quotient and remainder, we gain insights into how the dividend can be expressed in terms of the divisor, which is invaluable for various mathematical applications.

Method 1: Long Division of Polynomials

Long division of polynomials is a systematic approach analogous to the long division of numbers. It provides a clear, step-by-step method to find the quotient and remainder. Let's break down the process:

  1. Set up the division: Write the dividend (2x² + 7x - 15) inside the division symbol and the divisor (x + 5) outside.
  2. Divide the leading terms: Divide the leading term of the dividend (2x²) by the leading term of the divisor (x), which gives 2x. This is the first term of the quotient.
  3. Multiply the divisor by the quotient term: Multiply the entire divisor (x + 5) by 2x, resulting in 2x² + 10x.
  4. Subtract: Subtract the result (2x² + 10x) from the corresponding terms of the dividend (2x² + 7x), yielding -3x.
  5. Bring down the next term: Bring down the next term from the dividend (-15) to form the new expression -3x - 15.
  6. Repeat: Divide the leading term of the new expression (-3x) by the leading term of the divisor (x), which gives -3. This is the next term of the quotient.
  7. Multiply the divisor by the quotient term: Multiply the divisor (x + 5) by -3, resulting in -3x - 15.
  8. Subtract: Subtract the result (-3x - 15) from the current expression (-3x - 15), yielding 0.
  9. Interpret the result: The quotient is 2x - 3, and the remainder is 0. This means that (2x² + 7x - 15) is perfectly divisible by (x + 5).

Long division, while sometimes perceived as lengthy, offers a structured approach that is particularly useful for divisors with higher degrees. The step-by-step nature of the process minimizes the chances of errors and provides a clear visual representation of the division process. Furthermore, long division emphasizes the underlying principles of polynomial division, reinforcing the connection between the dividend, divisor, quotient, and remainder.

Method 2: Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear divisor of the form x - c. It's a faster and more compact alternative to long division, but it's applicable only when the divisor is linear. Let's apply it to our problem:

  1. Identify 'c': In the divisor x + 5, c is -5 (since x + 5 = x - (-5)).
  2. Set up the synthetic division: Write the coefficients of the dividend (2, 7, -15) in a row. Write 'c' (-5) to the left.
  3. Bring down the first coefficient: Bring down the first coefficient (2) to the bottom row.
  4. Multiply and add: Multiply the number you just brought down (2) by 'c' (-5), which gives -10. Write this under the next coefficient (7) and add them: 7 + (-10) = -3.
  5. Repeat: Multiply the result (-3) by 'c' (-5), which gives 15. Write this under the last coefficient (-15) and add them: -15 + 15 = 0.
  6. Interpret the result: The numbers in the bottom row (excluding the last one) are the coefficients of the quotient. The last number is the remainder. In this case, the quotient is 2x - 3, and the remainder is 0.

Synthetic division provides a highly efficient method for polynomial division when dealing with linear divisors. Its compactness and reduced number of steps make it a favorite among students and mathematicians alike. However, it's crucial to remember that synthetic division is strictly applicable to linear divisors. Attempting to use it with higher-degree divisors will lead to incorrect results. The elegance of synthetic division lies in its ability to distill the core operations of polynomial division into a simple, easily repeatable process. This makes it a valuable tool for quickly solving division problems and for building a deeper understanding of the relationships between polynomials.

The Solution: Quotient and Remainder

Both long division and synthetic division methods lead to the same result: the quotient is 2x - 3, and the remainder is 0. This means that (2x² + 7x - 15) ÷ (x + 5) = 2x - 3. In the context of the given multiple-choice options, the correct answer is A. 2x - 3. The zero remainder indicates that (x + 5) is a factor of (2x² + 7x - 15), meaning that the quadratic polynomial can be perfectly factored as (x + 5)(2x - 3). This connection between polynomial division and factorization is a fundamental concept in algebra, and mastering division helps in understanding and applying factorization techniques. The quotient, 2x - 3, represents the other factor of the quadratic polynomial, providing a complete picture of its structure and allowing for further analysis, such as finding its roots or sketching its graph.

Conclusion: Mastering Polynomial Division

Polynomial division, whether tackled through long division or synthetic division, is a vital skill in mathematics. It empowers us to simplify expressions, solve equations, and gain deeper insights into the relationships between polynomials. By understanding both methods, you equip yourself with versatile tools for tackling a wide range of mathematical problems. The problem (2x² + 7x - 15) ÷ (x + 5) serves as an excellent example to illustrate these techniques. Consistent practice and a firm grasp of the underlying principles will solidify your understanding and proficiency in polynomial division. Mastering polynomial division opens doors to advanced mathematical concepts and problem-solving strategies. It's a skill that pays dividends in various areas of mathematics and its applications, from calculus to engineering. The ability to confidently manipulate polynomial expressions is a hallmark of a strong mathematical foundation, and polynomial division is a crucial stepping stone on that path.

By practicing various polynomial division problems and understanding the nuances of each method, you can develop a strong foundation in algebra and excel in your mathematical journey. Remember, the key to success lies in consistent practice and a willingness to explore different approaches to problem-solving. So, embrace the challenge of polynomial division, and unlock its power to simplify and conquer complex mathematical expressions.