Importance of Grouping Like Terms

Oct 13, 2025 by vinooalluri.com 28 views

Polynomials are algebraic 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. In this article, we'll explore the concept of grouping like terms and determine which expression shows the sum of the polynomials with like terms grouped together.

Understanding Like Terms

Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1. Similarly, 3y^2 and 2y^2 are like terms because they both have the variable y raised to the power of 2.

Grouping Like Terms

When simplifying polynomials, we group like terms together by combining their coefficients. For instance, if we have the polynomial 2x + 4x, we can combine the like terms to get 6x. This process is called grouping like terms.

Evaluating the Options

Now, let's evaluate the given options to determine which one shows the sum of the polynomials with like terms grouped together.

Option A

${$ \left(8x + 3z - 8z^2\right) + \left(4y - 5z\right)$}$

In this option, the like terms are not grouped together. The terms 8x and 4y are not combined, and the terms 3z and -5z are not combined either.

Option B

${$ \left[8x + 4y + \left(-8z^2\right)\right] + 3z + (-5z)$}$

In this option, the like terms are partially grouped together. The terms 8x and 4y are combined, but the terms 3z and -5z are not combined.

Option C

${ $8x + 4y + \left[3z - 5z\right]\$}$

In this option, the like terms are fully grouped together. The terms 3z and -5z are combined to get -2z, and the terms 8x and 4y are combined as is.

Conclusion

Based on the evaluation of the options, we can conclude that the expression that shows the sum of the polynomials with like terms grouped together is:

${ $8x + 4y + \left[3z - 5z\right]\$}$

This expression fully groups the like terms together, making it the correct answer.

Importance of Grouping Like Terms

Grouping like terms is an essential step in simplifying polynomials. It helps to reduce the complexity of the polynomial and makes it easier to work with. By combining like terms, we can simplify the polynomial and make it more manageable.

Real-World Applications

Grouping like terms has numerous real-world applications in various fields, including physics, engineering, and economics. In physics, for example, grouping like terms is used to simplify complex equations that describe the motion of objects. In engineering, it's used to design and optimize systems, such as electrical circuits and mechanical systems. In economics, it's used to model and analyze complex economic systems.

Tips for Grouping Like Terms

Here are some tips for grouping like terms:

Identify the like terms in the polynomial.
Combine the coefficients of the like terms.
Simplify the resulting expression.

By following these tips, you can effectively group like terms and simplify polynomials.

Common Mistakes to Avoid

When grouping like terms, it's essential to avoid common mistakes, such as:

Not identifying like terms.
Not combining coefficients.
Not simplifying the resulting expression.

By avoiding these mistakes, you can ensure that your polynomial is simplified correctly.

Conclusion

In conclusion, grouping like terms is an essential step in simplifying polynomials. By combining like terms, we can reduce the complexity of the polynomial and make it easier to work with. In this article, we evaluated the given options and determined that the expression that shows the sum of the polynomials with like terms grouped together is:

${ $8x + 4y + \left[3z - 5z\right]\$}$

In our previous article, we discussed the importance of grouping like terms in simplifying polynomials. We also evaluated the given options and determined that the expression that shows the sum of the polynomials with like terms grouped together is:

${ $8x + 4y + \left[3z - 5z\right]\$}$

In this article, we'll answer some frequently asked questions about grouping like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: Why is it important to group like terms?

A: Grouping like terms is essential in simplifying polynomials. It helps to reduce the complexity of the polynomial and makes it easier to work with. By combining like terms, we can simplify the polynomial and make it more manageable.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, add or subtract their coefficients. For example, if we have the polynomial 2x + 4x, we can combine the like terms to get 6x.

Q: What are some common mistakes to avoid when grouping like terms?

A: Some common mistakes to avoid when grouping like terms include:

Not identifying like terms.
Not combining coefficients.
Not simplifying the resulting expression.

Q: How do I simplify a polynomial with like terms?

A: To simplify a polynomial with like terms, follow these steps:

Identify the like terms in the polynomial.
Combine the coefficients of the like terms.
Simplify the resulting expression.

Q: Can you give an example of a polynomial with like terms?

A: Here's an example of a polynomial with like terms:

${ $2x + 4x + 3y - 2y\$}$

In this example, the like terms are 2x and 4x, and 3y and -2y. We can combine the like terms to get:

${ $6x + y\$}$

Q: Can you give an example of a polynomial without like terms?

A: Here's an example of a polynomial without like terms:

${ $2x + 3y - 4z\$}$

In this example, there are no like terms because the variables x, y, and z are not raised to the same power.

Q: How do I know if a polynomial has like terms?

A: To determine if a polynomial has like terms, look for terms that have the same variable raised to the same power. If you find any, then the polynomial has like terms.

Q: Can you give a real-world example of grouping like terms?

A: Here's a real-world example of grouping like terms:

Suppose we have a polynomial that represents the cost of producing a certain product:

${ $2x + 3y - 4z\$}$

In this example, the polynomial has like terms because the variables x, y, and z are not raised to the same power. However, if we group the like terms, we get:

${ $2x + (3y - 4z)\$}$

This simplified polynomial is easier to work with and can be used to make decisions about production costs.

Conclusion

In conclusion, grouping like terms is an essential step in simplifying polynomials. By combining like terms, we can reduce the complexity of the polynomial and make it easier to work with. In this article, we answered some frequently asked questions about grouping like terms and provided examples to illustrate the concept. We hope this article has been helpful in understanding the importance of grouping like terms.