Which Expression Shows The Sum Of The Polynomials With Like Terms Grouped Together?$\[10x^2y + 2xy^2 - 4x^2 - 4x^2y\\]

Introduction

In algebra, polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. When dealing with polynomials, it's essential to simplify them by combining like terms. This process involves grouping together terms that have the same variable and exponent, making it easier to perform operations and understand the polynomial's structure.

What are Like Terms?

Like terms are terms in a polynomial that have the same variable and exponent. For example, in the polynomial $2x^2 + 3x^2$, the terms $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2. Similarly, in the polynomial $4y^3 - 2y^3$, the terms $4y^3$ and $-2y^3$ are like terms because they both have the variable $y$ raised to the power of 3.

Grouping Like Terms

To group like terms, we need to identify the terms in the polynomial that have the same variable and exponent. We can then combine these terms by adding or subtracting their coefficients. For example, in the polynomial $10x^2y + 2xy^2 - 4x^2 - 4x^2y$, we can group the like terms as follows:

• The terms $10x^2y$ and $-4x^2y$ are like terms because they both have the variable $x$ raised to the power of 2 and the variable $y$.
• The terms $2xy^2$ and $-4x^2$ are not like terms because they have different variables and exponents.

Expression Showing the Sum of Polynomials with Like Terms Grouped Together

To show the sum of the polynomials with like terms grouped together, we need to combine the like terms we identified earlier. We can do this by adding or subtracting the coefficients of the like terms. In this case, we have:

• $10x^2y + (-4x^2y) = 6x^2y$
• $2xy^2$ remains the same because it doesn't have any like terms.

Therefore, the expression showing the sum of the polynomials with like terms grouped together is:

6x^2y + 2xy^2

Example

Let's consider another example to illustrate the concept of grouping like terms. Suppose we have the polynomial $3x^2 + 2x^2 + 4x - 2x$. We can group the like terms as follows:

• The terms $3x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2.
• The terms $4x$ and $-2x$ are like terms because they both have the variable $x$.

We can then combine the like terms by adding or subtracting their coefficients. In this case, we have:

• $3x^2 + 2x^2 = 5x^2$
• $4x + (-2x) = 2x$

Therefore, the expression showing the sum of the polynomials with like terms grouped together is:

5x^2 + 2x

Conclusion

Q: What are like terms in a polynomial?

A: Like terms are terms in a polynomial that have the same variable and exponent. For example, in the polynomial $2x^2 + 3x^2$, the terms $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I identify like terms in a polynomial?

A: To identify like terms, you need to look for terms that have the same variable and exponent. You can do this by comparing the coefficients and variables of each term. For example, in the polynomial $4y^3 - 2y^3$, the terms $4y^3$ and $-2y^3$ are like terms because they both have the variable $y$ raised to the power of 3.

Q: How do I group like terms in a polynomial?

A: To group like terms, you need to combine the terms that have the same variable and exponent. You can do this by adding or subtracting the coefficients of the like terms. For example, in the polynomial $10x^2y + 2xy^2 - 4x^2 - 4x^2y$, the like terms are $10x^2y$ and $-4x^2y$, which can be combined as $6x^2y$.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents. For example, in the polynomial $4x^2 + 2y^2$, the terms $4x^2$ and $2y^2$ are unlike terms because they have different variables.

Q: Can I have a negative coefficient in a like term?

A: Yes, you can have a negative coefficient in a like term. For example, in the polynomial $-3x^2 + 2x^2$, the terms $-3x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I simplify a polynomial by grouping like terms?

A: To simplify a polynomial by grouping like terms, you need to identify the like terms and combine them by adding or subtracting their coefficients. For example, in the polynomial $3x^2 + 2x^2 + 4x - 2x$, the like terms are $3x^2$ and $2x^2$, which can be combined as $5x^2$, and the like terms are $4x$ and $-2x$, which can be combined as $2x$.

Q: Can I have a polynomial with no like terms?

A: Yes, you can have a polynomial with no like terms. For example, in the polynomial $2x^2 + 3y^2$, there are no like terms because the terms have different variables.

Q: How do I check if I have grouped like terms correctly?

A: To check if you have grouped like terms correctly, you need to make sure that you have combined all the like terms and that there are no terms left over. You can do this by re-reading the polynomial and making sure that each term has been combined with its like terms.

Conclusion

In conclusion, grouping like terms is an essential step in simplifying polynomials. By identifying and combining like terms, you can make it easier to perform operations and understand the polynomial's structure. We hope that this article has helped you to understand how to group like terms and simplify polynomials. If you have any further questions, please don't hesitate to ask.