Polynomial Practice Adding And Subtracting Expressions

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Welcome to a comprehensive guide on practicing the fundamental operations of adding and subtracting polynomial expressions. Polynomials, which are algebraic expressions consisting of variables and coefficients, are a cornerstone of mathematics. Mastering these operations is crucial for success in algebra and beyond. In this article, we'll delve into several examples, providing step-by-step solutions and explanations to solidify your understanding. Let's embark on this mathematical journey together!

1) (2a + 3b - c) + (5a - 6b + 8c)

In this initial problem, we're tasked with adding two polynomial expressions. The key to adding polynomials lies in combining like terms. Like terms are those that have the same variable raised to the same power. For instance, 2a and 5a are like terms, while 3b and -6b are also like terms, and -c and 8c are like terms as well. This concept of combining like terms is fundamental in polynomial arithmetic.

To solve this problem effectively, we need to methodically group the like terms together. This involves rearranging the expression to bring terms with the same variable and exponent next to each other. Once grouped, we can then add the coefficients of the like terms. The coefficient is the numerical part of the term; for example, in the term 2a, the coefficient is 2. Adding coefficients is a straightforward arithmetic operation, but it's crucial to pay attention to the signs (positive or negative) of the coefficients. A common mistake is overlooking a negative sign, which can lead to an incorrect answer. Therefore, careful attention to detail is essential in this step.

Let's break down the solution step by step:

  1. Identify like terms: In the expression (2a + 3b - c) + (5a - 6b + 8c), the like terms are 2a and 5a, 3b and -6b, and -c and 8c.
  2. Group like terms: Rearrange the expression to group these terms together: (2a + 5a) + (3b - 6b) + (-c + 8c).
  3. Add the coefficients: Now, add the coefficients of the like terms: (2 + 5)a + (3 - 6)b + (-1 + 8)c.
  4. Simplify: Perform the arithmetic operations: 7a - 3b + 7c.

Therefore, the sum of (2a + 3b - c) and (5a - 6b + 8c) is 7a - 3b + 7c. This result demonstrates the core principle of polynomial addition: combining like terms by adding their coefficients. The simplified expression represents the sum of the original polynomials in its most concise form. Understanding and applying this method correctly is essential for tackling more complex polynomial addition problems.

2) (4x² - 12x - 8) - (3x² + 9x - 5)

Moving on to the second problem, we encounter polynomial subtraction. Subtraction is very similar to addition, but with a crucial difference: we need to distribute the negative sign across the second polynomial. This means that each term inside the second set of parentheses will have its sign changed. Distributing the negative sign is a critical step in polynomial subtraction, and overlooking it is a common source of errors. It's like subtracting a group of items; you're removing each item from the group, which affects the sign of each item individually.

Once the negative sign is properly distributed, the problem transforms into an addition problem. We then proceed to combine like terms, just as we did in the previous example. This involves identifying terms with the same variable and exponent, grouping them together, and then adding their coefficients. The same principles of careful attention to signs and methodical grouping apply here as well.

Let's break down the solution process step-by-step:

  1. Distribute the negative sign: Rewrite the expression by distributing the negative sign to each term in the second polynomial: 4x² - 12x - 8 - 3x² - 9x + 5.
  2. Identify like terms: The like terms in this expression are 4x² and -3x², -12x and -9x, and -8 and 5.
  3. Group like terms: Rearrange the expression to group these terms together: (4x² - 3x²) + (-12x - 9x) + (-8 + 5).
  4. Add the coefficients: Add the coefficients of the like terms: (4 - 3)x² + (-12 - 9)x + (-8 + 5).
  5. Simplify: Perform the arithmetic operations: 1x² - 21x - 3, which can be written as x² - 21x - 3.

Therefore, the result of subtracting (3x² + 9x - 5) from (4x² - 12x - 8) is x² - 21x - 3. This process highlights the importance of correctly distributing the negative sign and then applying the rules of polynomial addition. The final simplified expression represents the difference between the two original polynomials.

3) (8a²b - 4ab²) + (12ab² - 4ab + 4ab²)

The third problem presents another addition of polynomial expressions, but this time we encounter terms with multiple variables. The principle of combining like terms remains the same, but we must be even more careful in identifying the terms that can be combined. Remember, like terms must have the same variables raised to the same powers. For example, a²b and ab² are not like terms because the exponents of a and b are different.

To approach this problem, we need to meticulously examine each term and identify its variable components and their exponents. This careful analysis is crucial for accurate grouping and subsequent addition. A common mistake is to combine terms that look similar but have different variable exponents. Therefore, a thorough understanding of what constitutes a like term is essential.

Let's solve this problem step-by-step:

  1. Identify like terms: In the expression (8a²b - 4ab²) + (12ab² - 4ab + 4ab²), the like terms are -4ab², 12ab², and 4ab². The term 8a²b and -4ab do not have any like terms to combine with.
  2. Group like terms: Rearrange the expression to group these terms together: 8a²b + (-4ab² + 12ab² + 4ab²) - 4ab.
  3. Add the coefficients: Add the coefficients of the like terms: 8a²b + (-4 + 12 + 4)ab² - 4ab.
  4. Simplify: Perform the arithmetic operations: 8a²b + 12ab² - 4ab.

Therefore, the sum of (8a²b - 4ab²) + (12ab² - 4ab + 4ab²) is 8a²b + 12ab² - 4ab. This solution emphasizes the importance of correctly identifying like terms when dealing with multiple variables. The final simplified expression represents the sum of the original polynomials, with only the like terms combined.

4) (5xy² - 6xy³ + 12xy) - (14xy - 12x²y² + 15xy³)

This fourth problem combines polynomial subtraction with terms containing multiple variables, making it a more complex challenge. As we learned in problem 2, the first step is to distribute the negative sign across the second polynomial. This is crucial for correctly accounting for the subtraction of each term. Neglecting to distribute the negative sign is a common mistake that can lead to an incorrect answer.

After distributing the negative sign, we proceed to identify and combine like terms. Again, careful attention to the variables and their exponents is essential. Terms must have the exact same variable components raised to the exact same powers in order to be considered like terms. This requires a meticulous examination of each term to ensure accurate grouping and combination.

Let's solve this problem with a step-by-step approach:

  1. Distribute the negative sign: Rewrite the expression by distributing the negative sign to each term in the second polynomial: 5xy² - 6xy³ + 12xy - 14xy + 12x²y² - 15xy³.
  2. Identify like terms: The like terms in this expression are -6xy³ and -15xy³, and 12xy and -14xy. The terms 5xy² and 12x²y² do not have any like terms.
  3. Group like terms: Rearrange the expression to group these terms together: 5xy² + (-6xy³ - 15xy³) + (12xy - 14xy) + 12x²y².
  4. Add the coefficients: Add the coefficients of the like terms: 5xy² + (-6 - 15)xy³ + (12 - 14)xy + 12x²y².
  5. Simplify: Perform the arithmetic operations: 5xy² - 21xy³ - 2xy + 12x²y².

Therefore, the result of subtracting (14xy - 12x²y² + 15xy³) from (5xy² - 6xy³ + 12xy) is 5xy² - 21xy³ - 2xy + 12x²y². This problem reinforces the importance of both distributing the negative sign correctly and carefully identifying like terms when dealing with multiple variables and exponents. The final simplified expression represents the difference between the two original polynomials.

5) 10x + (-5x - 2y) - (3x + 4y - 7)

In this final problem, we encounter a combination of addition and subtraction involving polynomials with two variables. This problem provides an excellent opportunity to consolidate our understanding of the principles we've discussed so far. We need to carefully handle the addition and subtraction operations, paying close attention to the signs of each term.

As with subtraction, we'll need to distribute the negative sign across the terms within the parentheses being subtracted. Then, we identify and combine like terms, ensuring that we only combine terms with the same variable and exponent. This methodical approach will help us arrive at the correct simplified expression.

Let's solve this problem step-by-step:

  1. Distribute the negative sign: Rewrite the expression by distributing the negative sign to each term in the last polynomial: 10x - 5x - 2y - 3x - 4y + 7.
  2. Identify like terms: The like terms in this expression are 10x, -5x, and -3x, and -2y and -4y. The constant term 7 does not have any like terms.
  3. Group like terms: Rearrange the expression to group these terms together: (10x - 5x - 3x) + (-2y - 4y) + 7.
  4. Add the coefficients: Add the coefficients of the like terms: (10 - 5 - 3)x + (-2 - 4)y + 7.
  5. Simplify: Perform the arithmetic operations: 2x - 6y + 7.

Therefore, the simplified form of the expression 10x + (-5x - 2y) - (3x + 4y - 7) is 2x - 6y + 7. This problem serves as a comprehensive review of the techniques we've covered in this article, including distributing the negative sign and combining like terms. The final simplified expression represents the result of the combined addition and subtraction operations.

Conclusion

In conclusion, mastering the addition and subtraction of polynomial expressions is a fundamental skill in algebra. By understanding the principles of combining like terms and carefully handling the distribution of negative signs, you can confidently tackle a wide range of polynomial problems. Remember to always pay close attention to the signs of the coefficients and to ensure that you are only combining terms with the same variable and exponent. With consistent practice, you'll develop the fluency and accuracy needed to excel in this crucial area of mathematics. Keep practicing, and you'll become a polynomial pro in no time!