Polynomial Subtraction Find The Difference Of Expressions

In the realm of algebra, manipulating polynomials is a fundamental skill. One common operation is polynomial subtraction, where we find the difference between two polynomial expressions. This process involves careful application of the distributive property and combining like terms. Let's embark on a journey to master the art of subtracting polynomials, using the expression $(4x^2y^3 + 2xy^2 - 2y) - (-7x^2y^3 + 6xy^2 - 2y)$ as our guiding example.

Polynomial subtraction might seem daunting at first, but it's simply the process of finding the difference between two polynomial expressions. The key to success lies in understanding the distributive property and the concept of combining like terms. Let's break down these concepts before diving into our example. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. In the context of polynomials, this means we can multiply a term outside parentheses by each term inside the parentheses. Like terms are terms that have the same variables raised to the same powers. For example, $3x^2y$ and $-5x^2y$ are like terms because they both have $x^2$ and $y$. However, $3x^2y$ and $3xy^2$ are not like terms because the exponents of x and y are different. To combine like terms, we simply add or subtract their coefficients. For instance, $3x^2y - 5x^2y = -2x^2y$. With these concepts in mind, we can confidently tackle polynomial subtraction.

Now, let's delve into our example: $(4x^2y^3 + 2xy^2 - 2y) - (-7x^2y^3 + 6xy^2 - 2y)$. The first step is to distribute the negative sign in front of the second polynomial. This is equivalent to multiplying each term inside the parentheses by -1. Doing so, we get: $4x^2y^3 + 2xy^2 - 2y + 7x^2y^3 - 6xy^2 + 2y$. Notice how the signs of each term in the second polynomial have changed. Next, we identify like terms. In this expression, we have two terms with $x^2y^3$, two terms with $xy^2$, and two terms with y. Now, we combine these like terms: $(4x^2y^3 + 7x^2y^3) + (2xy^2 - 6xy^2) + (-2y + 2y)$. Adding the coefficients of the like terms, we get: $11x^2y^3 - 4xy^2 + 0y$. Finally, we simplify the expression by removing the term with a coefficient of 0: $11x^2y^3 - 4xy^2$. Therefore, the difference between the two polynomials is $11x^2y^3 - 4xy^2$. By carefully applying the distributive property and combining like terms, we have successfully subtracted the polynomials. This example serves as a foundation for tackling more complex polynomial subtraction problems. Remember, the key is to break down the problem into smaller, manageable steps and to pay close attention to the signs of the terms.

Step-by-Step Solution

To accurately find the difference between polynomials, we need to meticulously follow a step-by-step approach. This involves distributing the negative sign, identifying like terms, and combining them correctly. Let's break down the given expression, $(4x^2y^3 + 2xy^2 - 2y) - (-7x^2y^3 + 6xy^2 - 2y)$, into manageable steps.

Our first crucial step in polynomial subtraction is to distribute the negative sign. This means multiplying each term within the second set of parentheses by -1. The original expression is $(4x^2y^3 + 2xy^2 - 2y) - (-7x^2y^3 + 6xy^2 - 2y)$. Distributing the negative sign, we get: $4x^2y^3 + 2xy^2 - 2y + 7x^2y^3 - 6xy^2 + 2y$. Notice how the signs of the terms inside the second set of parentheses have changed: -7x^2y3 becomes +7x^2y3, +6xy^2 becomes -6xy^2, and -2y becomes +2y. This step is crucial because it sets the stage for accurately combining like terms. Forgetting to distribute the negative sign will lead to an incorrect answer. Think of the subtraction as adding the opposite. Instead of subtracting the entire polynomial, we are adding the negative of the polynomial. This mental shift can help to avoid errors. Distributing the negative sign ensures that we are accounting for the correct signs of each term when we combine like terms in the next step. This step is like laying the foundation for a building; if the foundation is not solid, the entire structure will be weak. Therefore, take your time and double-check your work to ensure that you have correctly distributed the negative sign. By paying attention to detail in this step, you will significantly increase your chances of arriving at the correct solution. The next step involves identifying like terms, which we will discuss in detail in the following section.

Next, we must identify like terms within the expression. Like terms are those that have the same variables raised to the same powers. In our expression, $4x^2y^3 + 2xy^2 - 2y + 7x^2y^3 - 6xy^2 + 2y$, we can identify three pairs of like terms: $4x^2y^3$ and $7x^2y^3$ are like terms because they both have $x^2y^3$. $2xy^2$ and $-6xy^2$ are like terms because they both have $xy^2$. $-2y$ and $+2y$ are like terms because they both have $y$. Identifying like terms is like sorting objects into categories. We are grouping together the terms that are similar so that we can combine them in the next step. To effectively identify like terms, pay close attention to the variables and their exponents. The coefficients (the numbers in front of the variables) do not matter when identifying like terms. What matters is that the variables and their exponents are the same. For example, $5x^3$ and $-2x^3$ are like terms, but $5x^3$ and $-2x^2$ are not like terms because the exponents are different. Similarly, $3xy^2$ and $7xy^2$ are like terms, but $3xy^2$ and $7x^2y$ are not like terms because the exponents of x and y are different. A helpful strategy for identifying like terms is to use different colors or shapes to mark them. For example, you could circle all the $x^2y^3$ terms in red, underline all the $xy^2$ terms in blue, and put a box around all the y terms in green. This visual aid can make it easier to see which terms are like terms. Once you have identified the like terms, you are ready to combine them in the next step. Combining like terms is like adding up the objects in each category. We will add the coefficients of the like terms to simplify the expression.

Finally, we combine the like terms identified in the previous step. This involves adding or subtracting the coefficients of the like terms while keeping the variables and their exponents the same. From our expression, we have: $(4x^2y^3 + 7x^2y^3) + (2xy^2 - 6xy^2) + (-2y + 2y)$. Combining the coefficients, we get: $11x^2y^3 - 4xy^2 + 0y$. The final step is to simplify the expression by removing any terms with a coefficient of 0. In this case, we have $0y$, which is equal to 0, so we can remove it from the expression. This leaves us with the simplified expression: $11x^2y^3 - 4xy^2$. Therefore, the difference between the two original polynomials is $11x^2y^3 - 4xy^2$. Combining like terms is like putting the final touches on a painting. It's where we bring everything together and simplify the expression to its most basic form. When combining like terms, it's essential to pay attention to the signs of the coefficients. Adding a negative number is the same as subtracting a positive number, and subtracting a negative number is the same as adding a positive number. For example, $2xy^2 - 6xy^2$ is the same as $2xy^2 + (-6xy^2)$, which gives us $-4xy^2$. A common mistake is to change the exponents when combining like terms. Remember, we only add or subtract the coefficients; the variables and their exponents stay the same. For example, $4x^2y^3 + 7x^2y^3 = 11x^2y^3$, not $11x^4y^6$. After combining all the like terms, double-check your work to ensure that you have not made any errors. It's always a good idea to simplify the expression as much as possible. This means removing any terms with a coefficient of 0 and writing the terms in descending order of their exponents. By following these steps carefully, you can confidently subtract polynomials and arrive at the correct solution.

Filling in the Coefficients

Now, let's apply our understanding to the specific question of placing the correct coefficients in the difference. We have determined that the difference between $(4x^2y^3 + 2xy^2 - 2y)$ and $(-7x^2y^3 + 6xy^2 - 2y)$ is $11x^2y^3 - 4xy^2$. Therefore, we can directly identify the coefficients for each term.

The question asks us to fill in the blanks in the expression: oxed{ } x^2y^3 + oxed{ } xy^2 + oxed{ } y. We have already found that the difference between the two polynomials is $11x^2y^3 - 4xy^2$. Comparing this to the given expression, we can see that: The coefficient of $x^2y^3$ is 11. The coefficient of $xy^2$ is -4. The coefficient of $y$ is 0 (since there is no y term in the simplified expression). Therefore, the correct coefficients to fill in the blanks are 11, -4, and 0. This exercise highlights the importance of carefully following the steps of polynomial subtraction. By distributing the negative sign, identifying like terms, and combining them correctly, we were able to arrive at the simplified expression and easily identify the coefficients. Filling in the coefficients is like putting the final pieces of a puzzle together. It's where we take the simplified expression and extract the specific information that the question is asking for. In this case, the question was asking for the coefficients of each term, which we were able to identify by comparing the simplified expression to the given form. A common mistake is to forget the negative sign when identifying the coefficients. For example, the coefficient of $xy^2$ in the expression $11x^2y^3 - 4xy^2$ is -4, not 4. Always pay close attention to the signs of the terms when identifying the coefficients. Another important point to remember is that if a term is missing in the simplified expression, its coefficient is 0. For example, in our case, the term y is missing, so its coefficient is 0. By carefully identifying the coefficients and paying attention to the signs and missing terms, you can confidently answer questions that ask you to fill in the coefficients in a polynomial expression.

Therefore, the final answer is:

oxed{11} $x^2y^3$ + oxed{-4} $xy^2$ + oxed{0} y

Conclusion: Mastering Polynomial Subtraction

In conclusion, subtracting polynomials is a fundamental algebraic skill that requires careful attention to detail. By understanding the distributive property and the process of combining like terms, we can confidently tackle any polynomial subtraction problem. Our example, $(4x^2y^3 + 2xy^2 - 2y) - (-7x^2y^3 + 6xy^2 - 2y)$, demonstrates the step-by-step approach to solving these problems.

Mastering polynomial subtraction is like learning to ride a bicycle. At first, it may seem challenging and require a lot of focus and effort. But with practice and persistence, it becomes second nature. The key is to break down the process into smaller, manageable steps and to understand the underlying principles. By following the steps outlined in this guide, you can develop a solid foundation in polynomial subtraction and build your confidence in algebra. Remember, the distributive property is your friend. It allows you to remove the parentheses and set the stage for combining like terms. Identifying like terms is like sorting your socks. You group together the items that are similar so that you can work with them more efficiently. Combining like terms is like adding up the numbers in your bank account. You add the positive numbers and subtract the negative numbers to find your balance. Polynomial subtraction is not just a mathematical skill; it's also a valuable life skill. It teaches you to pay attention to detail, to follow a systematic process, and to solve problems step by step. These skills are applicable in many areas of life, from managing your finances to planning a project to making important decisions. So, embrace the challenge of polynomial subtraction and take pride in your ability to master this important skill. With practice and dedication, you will become a confident and proficient algebra student.

We began by distributing the negative sign, effectively changing the subtraction problem into an addition problem. Then, we meticulously identified like terms, grouping together terms with the same variables and exponents. Finally, we combined these like terms by adding their coefficients, resulting in the simplified expression $11x^2y^3 - 4xy^2$. This process allowed us to accurately determine the coefficients for the given expression, filling in the blanks with 11, -4, and 0.

This comprehensive guide has provided you with the tools and knowledge necessary to confidently approach polynomial subtraction problems. Remember to practice regularly, and don't hesitate to break down complex problems into smaller, more manageable steps. With dedication and a solid understanding of the fundamentals, you'll excel in your algebraic endeavors.

Polynomial subtraction, distributive property, like terms, coefficients, algebraic expressions, step-by-step solution, simplify, variables, exponents