What Is The Result Of Subtracting The Polynomial (-2xy + 3) From The Polynomial (7y^2 + 6xy)?

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Polynomial subtraction might seem daunting at first, but it becomes manageable with a clear understanding of the underlying principles. This article delves into the process of subtracting polynomials, using a specific example to illustrate the steps involved. Our main focus will be on unraveling the nuances of polynomial subtraction, highlighting common pitfalls, and ensuring you grasp the concept thoroughly. Let’s embark on this mathematical journey together, making polynomial subtraction less of a hurdle and more of an enjoyable challenge.

Breaking Down Polynomial Subtraction

Polynomials, at their core, are algebraic expressions comprising variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Understanding polynomial subtraction requires a solid foundation in basic algebraic principles. When we subtract one polynomial from another, we are essentially finding the difference between their terms. This involves distributing the negative sign across the terms of the polynomial being subtracted and then combining like terms. Like terms are those that have the same variables raised to the same powers. For instance, 3x^2 and 5x^2 are like terms, while 3x^2 and 5x^3 are not. The coefficients of like terms can be added or subtracted, but the variables and their exponents remain unchanged. This concept is crucial in simplifying polynomial expressions. Mastering the art of identifying and combining like terms is fundamental to performing polynomial operations, including addition, subtraction, multiplication, and division. When subtracting polynomials, it is beneficial to rewrite the subtraction as addition of the negative of the second polynomial. This transformation often simplifies the process and reduces the chance of making errors. Precision in applying the distributive property and correctly identifying like terms are key to success in polynomial subtraction. Attention to detail in each step of the process is essential for obtaining the correct result. Polynomial subtraction is not just a mathematical exercise; it is a fundamental skill that finds applications in various fields, including engineering, physics, and computer science. From modeling physical phenomena to solving complex equations, the ability to manipulate polynomials is a valuable asset. Regular practice and a clear understanding of the rules governing polynomial operations will pave the way for proficiency in this area of mathematics.

Example: Subtracting Polynomials Step-by-Step

Let's illustrate polynomial subtraction with the example given:

(7y² + 6xy) - (-2xy + 3)

This example provides a practical scenario for understanding the mechanics of polynomial subtraction. We will dissect this problem step-by-step, elucidating the rationale behind each operation. The first key step in solving this problem is to distribute the negative sign across the terms within the second parentheses. This is crucial because it effectively changes the signs of the terms being subtracted. When we distribute the negative sign, the expression -(-2xy + 3) becomes +2xy - 3. This transformation is essential for correctly combining like terms in the subsequent steps. It's important to remember that distributing the negative sign affects every term within the parentheses, not just the first one. Neglecting to distribute the negative sign properly is a common error that can lead to an incorrect answer. Once the negative sign has been correctly distributed, the expression now reads 7y² + 6xy + 2xy - 3. We are now in a position to identify and combine like terms. Like terms, as previously discussed, are terms that have the same variables raised to the same powers. In this expression, 6xy and 2xy are like terms. The term 7y² is unique in that it has no other y² term to combine with, and the constant term -3 also stands alone. The next step is to combine the coefficients of the like terms 6xy and 2xy. Adding the coefficients 6 and 2 gives us 8. Therefore, 6xy + 2xy simplifies to 8xy. Now we can rewrite the entire expression by combining the like terms. The expression 7y² + 6xy + 2xy - 3 simplifies to 7y² + 8xy - 3. This final expression represents the difference between the two original polynomials. It is in its simplest form, as there are no more like terms to combine. The process of simplifying polynomial expressions is a fundamental skill in algebra, and this example illustrates the key steps involved. Understanding how to distribute the negative sign and combine like terms is essential for mastering polynomial subtraction.

Step-by-Step Solution

  1. Distribute the negative sign: This is a crucial first step in polynomial subtraction. The given expression is

    (7y² + 6xy) - (-2xy + 3)

    Distributing the negative sign across the second polynomial changes the signs of its terms:

    7y² + 6xy + 2xy - 3

    Distributing the negative sign is akin to multiplying the second polynomial by -1. This operation effectively reverses the sign of each term within the parentheses. For example, -(-2xy) becomes +2xy, and -(+3) becomes -3. This transformation is vital for the subsequent step of combining like terms. Without proper distribution of the negative sign, the resulting expression will be incorrect. The concept of distribution is a fundamental principle in algebra, and it applies not only to polynomial subtraction but also to other operations such as multiplication. It is essential to understand that the negative sign in front of the parentheses applies to the entire polynomial within the parentheses, not just the first term. This is why each term's sign must be changed. After distributing the negative sign, the expression is now in a form where like terms can be easily identified and combined. This step sets the stage for simplifying the polynomial expression to its simplest form. Accuracy in this step is paramount, as any error in sign distribution will propagate through the rest of the solution. It is often helpful to rewrite the subtraction as addition of the negative, making the distribution process more intuitive.

  2. Combine like terms: Identify and combine terms with the same variables and exponents. In this case, 6xy and 2xy are like terms. So, we can combine polynomial subtraction these:

    7y² + (6xy + 2xy) - 3

    7y² + 8xy - 3

    Combining like terms is a fundamental step in simplifying polynomial expressions. It involves adding or subtracting the coefficients of terms that have the same variable(s) raised to the same power(s). In our example, 6xy and 2xy are like terms because they both have the variables x and y raised to the power of 1. The coefficients of these terms are 6 and 2, respectively. To combine them, we simply add the coefficients: 6 + 2 = 8. Therefore, 6xy + 2xy simplifies to 8xy. The other terms in the expression, 7y² and -3, do not have any like terms to combine with. The term 7y² has the variable y raised to the power of 2, and there are no other terms with the same variable and exponent. Similarly, -3 is a constant term, and there are no other constant terms in the expression. When combining like terms, it is crucial to pay attention to the signs of the coefficients. If we were subtracting terms, we would subtract the coefficients instead of adding them. Combining like terms is an application of the distributive property in reverse. The distributive property states that a(b + c) = ab + ac. In the case of combining like terms, we are essentially factoring out the common variable(s) and exponent(s). For example, 6xy + 2xy can be thought of as (6 + 2)xy, which simplifies to 8xy. Combining like terms is not only essential for simplifying polynomial expressions but also for solving equations and performing other algebraic operations. It allows us to reduce the complexity of expressions and make them easier to work with.

Final Answer

Therefore, the difference between the two polynomials is:

7y² + 8xy - 3

The correct answer is B. 7y² + 8xy - 3.

Common Mistakes to Avoid in Polynomial Subtraction

Mastering polynomial subtraction involves understanding the core principles and avoiding common pitfalls. Here are some mistakes to watch out for:

  1. Incorrectly Distributing the Negative Sign: This is the most prevalent mistake in polynomial subtraction. Remember, the negative sign in front of the parentheses applies to every term inside the parentheses. For example:

    (4x² - 3x + 2) - (x² + 2x - 1)

    The negative sign must be distributed to each term in the second polynomial:

    4x² - 3x + 2 - x² - 2x + 1

    Failing to distribute the negative sign correctly can lead to a completely different result. It is crucial to pay close attention to the signs of each term when distributing. A helpful strategy is to rewrite the subtraction as addition of the negative. For instance, the above example can be rewritten as (4x² - 3x + 2) + (-1)(x² + 2x - 1). This makes it more explicit that each term inside the second parentheses must be multiplied by -1. Another common error is distributing the negative sign only to the first term inside the parentheses. This is incorrect and will lead to an inaccurate answer. The negative sign acts as a multiplier for the entire polynomial, so every term must be affected. To avoid this mistake, it is beneficial to visualize the distribution process and mentally check that the sign of each term has been correctly changed. Regular practice with polynomial subtraction problems will help reinforce the correct application of the distributive property.

  2. Combining Non-Like Terms: Like terms have the same variable(s) raised to the same power(s). For example, 3x² and 5x² are like terms, but 3x² and 5x are not. Avoid combining terms that are not alike.

    2y³ + 4y² - y + 7 - (y³ - 2y² + 3y - 2)

    Distribute the negative sign:

    2y³ + 4y² - y + 7 - y³ + 2y² - 3y + 2

    Combine like terms:

    (2y³ - y³) + (4y² + 2y²) + (-y - 3y) + (7 + 2)

    y³ + 6y² - 4y + 9

    Combining non-like terms is a fundamental error in polynomial arithmetic. It stems from a misunderstanding of what constitutes a like term. Like terms, as previously mentioned, must have the same variable(s) raised to the same power(s). For example, x² and x are not like terms because the variable x is raised to different powers. Similarly, xy and x are not like terms because they do not have the same combination of variables. To avoid combining non-like terms, it is crucial to carefully examine the variables and their exponents before attempting to add or subtract coefficients. A helpful strategy is to group like terms together before performing any operations. This visual organization can make it easier to identify which terms can be combined. Another common mistake is to combine terms that have the same variable but different coefficients. While these terms are like terms and can be combined, it is essential to add or subtract the coefficients correctly. For example, 3x + 5x combines to 8x, not 8x². Regular practice with polynomial simplification will help solidify the understanding of like terms and reduce the likelihood of making this error. Paying close attention to detail and double-checking the variables and exponents before combining terms are essential for accuracy in polynomial arithmetic.

  3. Sign Errors: Be careful with signs, especially when distributing a negative sign or combining terms with negative coefficients.

    5a² - 3ab + 2b² - (2a² + ab - b²)

    Distribute the negative sign:

    5a² - 3ab + 2b² - 2a² - ab + b²

    Combine like terms:

    (5a² - 2a²) + (-3ab - ab) + (2b² + b²)

    3a² - 4ab + 3b²

Sign errors are a pervasive issue in algebra, and polynomial subtraction is no exception. These errors typically arise when distributing the negative sign or when combining terms with negative coefficients. As previously emphasized, the negative sign in front of the parentheses must be distributed to every term inside the parentheses. Failing to do so or changing the sign of only some terms will lead to an incorrect result. Another common source of sign errors is when combining like terms with negative coefficients. It is essential to remember the rules of integer arithmetic when adding or subtracting negative numbers. For example, -3x - 2x equals -5x, not -x. To minimize sign errors, it is helpful to rewrite subtraction as addition of the negative, as this can make the distribution process more transparent. Additionally, taking the time to double-check the signs of each term before combining them can prevent mistakes. A useful strategy is to circle like terms with the same sign and box like terms with the opposite sign. This visual cue can help ensure that the correct operation is performed. Regular practice and a methodical approach are key to reducing sign errors in polynomial subtraction. Paying close attention to detail and taking the time to verify each step of the process will significantly improve accuracy.

Practice Problems for Polynomial Subtraction

To solidify your understanding of polynomial subtraction, practice is key. Try these problems:

  1. (3x² + 2x - 1) - (x² - x + 4)
  2. (5y³ - 2y + 3) - (2y³ + y² - y)
  3. (4a²b - 3ab² + 2) - (2a²b + ab² - 1)

Working through these practice problems will reinforce your understanding of polynomial subtraction and help you develop confidence in your ability to solve such problems. Remember to focus on distributing the negative sign correctly and combining like terms accurately. Check your answers against the solutions provided below to ensure you are on the right track. The more you practice, the more proficient you will become in polynomial arithmetic.

Solutions to Practice Problems

**(3x² + 2x - 1) - (x² - x + 4)**

3x² + 2x - 1 - x² + x - 4


(3x² - x²) + (2x + x) + (-1 - 4)


2x² + 3x - 5


**(5y³ - 2y + 3) - (2y³ + y² - y)**

5y³ - 2y + 3 - 2y³ - y² + y


(5y³ - 2y³) - y² + (-2y + y) + 3


3y³ - y² - y + 3


**(4a²b - 3ab² + 2) - (2a²b + ab² - 1)**

4a²b - 3ab² + 2 - 2a²b - ab² + 1


(4a²b - 2a²b) + (-3ab² - ab²) + (2 + 1)


2a²b - 4ab² + 3

Conclusion

Polynomial subtraction, while seemingly complex, is a systematic process that becomes easier with practice. By understanding the steps involved, avoiding common mistakes, and working through practice problems, you can master this essential algebraic skill. Remember, the key is to distribute the negative sign correctly and combine like terms accurately. This skill is not only crucial for success in mathematics but also provides a foundation for more advanced concepts in various fields. With dedication and consistent practice, you'll be well-equipped to tackle any polynomial subtraction problem that comes your way.