Grouping Like Terms In Polynomial Expressions A Step By Step Guide

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Understanding Polynomial Expressions and Combining Like Terms

In the realm of algebra, polynomial expressions serve as fundamental building blocks for more complex mathematical models. A polynomial is essentially an expression comprising variables and coefficients, combined using the operations of addition, subtraction, and multiplication. The variables within a polynomial can be raised to non-negative integer powers, creating terms such as constants, linear terms, quadratic terms, and so on. When dealing with polynomials, a crucial skill is the ability to simplify them by combining like terms. Like terms are those that share the same variable(s) raised to the same power(s). For instance, in the polynomial expression provided, $10x^2y$ and $-4x^2y$ are like terms because they both involve the variables x and y raised to the powers of 2 and 1, respectively. The process of grouping like terms together allows us to consolidate the expression, making it easier to understand, manipulate, and ultimately solve.

To effectively combine like terms, we rely on the distributive property of multiplication over addition and subtraction. This property states that for any numbers a, b, and c, the equation a(b + c) = ab + ac holds true. In the context of polynomials, this means that we can add or subtract the coefficients of like terms while keeping the variable part unchanged. For example, to combine $10x^2y$ and $-4x^2y$, we add their coefficients (10 and -4) to get 6, resulting in the simplified term $6x^2y$. By systematically grouping and combining like terms, we can transform complex polynomial expressions into their simplest and most manageable forms. This process not only simplifies calculations but also provides valuable insights into the structure and behavior of the polynomial.

Analyzing the Given Polynomial Expression

Now, let's delve into the specific polynomial expression at hand: $10x^2y + 2xy^2 - 4x^2 - 4x^2y$. Our objective is to identify the expression that correctly groups the like terms together. Before we examine the provided options, it's essential to pinpoint the like terms within this polynomial. As we discussed earlier, like terms possess the same variable(s) raised to the same power(s). In this case, we have the following terms:

  • 10x^2y$: This term has the variable x raised to the power of 2 and the variable y raised to the power of 1.

  • 2xy^2$: This term has the variable x raised to the power of 1 and the variable y raised to the power of 2. Notice that this term is not a like term with $10x^2y$ because the powers of x and y are different.

  • -4x^2$: This term has the variable x raised to the power of 2, and it does not contain the variable y.

  • -4x^2y$: This term has the variable x raised to the power of 2 and the variable y raised to the power of 1. This term is a like term with $10x^2y$ because they both have the same variables raised to the same powers.

From this analysis, we can identify that $10x^2y$ and $-4x^2y$ are like terms. The other terms, $2xy^2$ and $-4x^2$, do not have any like terms within the given expression. Therefore, when we group the like terms together, we should have the terms $10x^2y$ and $-4x^2y$ grouped in the same set of parentheses or brackets. This grouping will allow us to combine their coefficients and simplify the expression further. Now that we have a clear understanding of the like terms within the polynomial, we can evaluate the provided options and determine which one accurately represents the sum of the polynomials with like terms grouped together.

Evaluating the Options

Now, let's examine the provided options to determine which one correctly groups the like terms together in the given polynomial expression:

Option A: $[(-4x^2) + (-4x^2y) + 10x^2y] + 2xy^2$

This option groups the terms $-4x^2$, $-4x^2y$, and $10x^2y$ together within the brackets. Let's analyze this grouping. As we identified earlier, $10x^2y$ and $-4x^2y$ are like terms because they both have the same variables (x and y) raised to the same powers (2 and 1, respectively). However, the term $-4x^2$ is not a like term with the other two because it only contains the variable x raised to the power of 2 and does not have the variable y. Therefore, this grouping is not entirely accurate as it includes a term that is not a like term with the others.

However, it does correctly group the like terms $-4x^2y$ and $10x^2y$ together. The term $2xy^2$ is kept separate, which is also correct since it does not have any like terms in the given expression. While the grouping is not perfect due to the inclusion of $-4x^2$, it does show an attempt to group like terms together.

Option B: $10x^2y + 2xy^2 - 4x^2 - 4x^2y$

This option simply restates the original polynomial expression without any grouping of like terms. It does not use any parentheses or brackets to visually separate or group the like terms together. Therefore, this option does not fulfill the requirement of showing the sum of the polynomials with like terms grouped together. It is just the original expression in its initial form.

To determine the correct answer, we need to identify the option that explicitly groups the like terms $10x^2y$ and $-4x^2y$ together, while keeping the unlike terms separate. Option A makes an attempt to group like terms but includes an unlike term in the grouping. Option B does not group any terms at all.

Determining the Correct Expression

Based on our analysis of the given polynomial expression and the provided options, we can now determine the correct expression that shows the sum of the polynomials with like terms grouped together. Recall that the polynomial expression is: $10x^2y + 2xy^2 - 4x^2 - 4x^2y$. We identified that the like terms in this expression are $10x^2y$ and $-4x^2y$. The other terms, $2xy^2$ and $-4x^2$, do not have any like terms within the expression.

Therefore, the correct expression should group $10x^2y$ and $-4x^2y$ together, while keeping $2xy^2$ and $-4x^2$ separate. Let's revisit the options:

  • Option A: $[(-4x^2) + (-4x^2y) + 10x^2y] + 2xy^2$ This option groups $-4x^2$, $-4x^2y$, and $10x^2y$ together. While it correctly groups $10x^2y$ and $-4x^2y$, it also includes $-4x^2$, which is not a like term. Therefore, this option is not entirely correct.
  • Option B: $10x^2y + 2xy^2 - 4x^2 - 4x^2y$ This option does not group any terms together and simply restates the original expression. Therefore, this option is not correct.

To create the correct expression, we need to group the like terms $10x^2y$ and $-4x^2y$ together using parentheses or brackets. The other terms should remain separate. The correct expression would look like this: $(10x^2y - 4x^2y) + 2xy^2 - 4x^2$. This expression clearly shows the like terms grouped together, allowing for simplification by combining their coefficients. By combining the like terms, we get $6x^2y + 2xy^2 - 4x^2$, which is the simplified form of the original polynomial expression.

Final Answer

In conclusion, to determine which expression shows the sum of the polynomials with like terms grouped together, we must identify the like terms within the polynomial and then group them appropriately using parentheses or brackets. In the given polynomial expression, $10x^2y + 2xy^2 - 4x^2 - 4x^2y$, the like terms are $10x^2y$ and $-4x^2y$.

After evaluating the provided options, we found that none of the options perfectly represent the sum of the polynomials with like terms grouped together. Option A attempts to group like terms but includes an unlike term in the grouping. Option B simply restates the original expression without any grouping.

The correct expression should group the like terms $10x^2y$ and $-4x^2y$ together, while keeping the unlike terms $2xy^2$ and $-4x^2$ separate. The correct expression would be: $(10x^2y - 4x^2y) + 2xy^2 - 4x^2$. This expression clearly shows the like terms grouped together, allowing for simplification.