Which Expression Shows The Sum Of The Polynomials With Like Terms Grouped Together?A. 10 X 2 Y + 2 X Y 2 − 4 X 2 − 4 X 2 Y 10x^2y + 2xy^2 - 4x^2 - 4x^2y 10 X 2 Y + 2 X Y 2 − 4 X 2 − 4 X 2 Y B. [ ( − 4 X 2 ) + ( − 4 X 2 Y ) + 10 X 2 Y ] + 2 X Y 2 \left[(-4x^2) + (-4x^2y) + 10x^2y\right] + 2xy^2 [ ( − 4 X 2 ) + ( − 4 X 2 Y ) + 10 X 2 Y ] + 2 X Y 2 C. $10x^2y + 2xy^2 + \left[(-4x^2) + (-4x^2y)\right]D.

Introduction

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. When simplifying polynomials, it is essential to group like terms together to make the expression more manageable and easier to work with. In this article, we will explore the concept of grouping like terms and determine which expression shows the sum of the polynomials with like terms grouped together.

What are Like Terms?

Like terms are terms in a polynomial that have the same variable(s) raised to the same power. 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. 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

Grouping like terms involves combining the coefficients of like terms to simplify the polynomial. For example, in the polynomial $3x^2 + 2x^2 + 4x^2$, we can group the like terms together as follows:

$$3x^2 + 2x^2 + 4x^2 = (3 + 2 + 4)x^2 = 9x^2$$

In this example, we combined the coefficients of the like terms $3x^2$, $2x^2$, and $4x^2$ to get the simplified expression $9x^2$.

Evaluating the Options

Now that we have a good understanding of like terms and grouping, let's evaluate the options given in the problem.

Option A

$10x^2y + 2xy^2 - 4x^2 - 4x^2y$

In this option, the like terms are not grouped together. The terms $-4x^2$ and $-4x^2y$ are like terms, but they are not combined.

Option B

$\left[(-4x^2) + (-4x^2y) + 10x^2y\right] + 2xy^2$

In this option, the like terms are grouped together. The terms $-4x^2$, $-4x^2y$, and $10x^2y$ are like terms, and they are combined as follows:

$$(-4x^2) + (-4x^2y) + 10x^2y = -4x^2 - 4x^2y + 10x^2y = 6x^2y - 4x^2$$

So, the expression becomes:

$$6x^2y - 4x^2 + 2xy^2$$

Option C

$10x^2y + 2xy^2 + \left[(-4x^2) + (-4x^2y)\right]$

In this option, the like terms are not grouped together. The terms $-4x^2$ and $-4x^2y$ are like terms, but they are not combined.

Option D

This option is not provided.

Conclusion

Based on our analysis, the correct answer is Option B. The expression $\left[(-4x^2) + (-4x^2y) + 10x^2y\right] + 2xy^2$ shows the sum of the polynomials with like terms grouped together.

Final Answer

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Q&A: Simplifying Polynomials with Like Terms

Q: What are like terms in a polynomial?

A: Like terms are terms in a polynomial that have the same variable(s) raised to the same power. 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 group like terms in a polynomial?

A: To group like terms, combine the coefficients of like terms. For example, in the polynomial $3x^2 + 2x^2 + 4x^2$, we can group the like terms together as follows:

$$3x^2 + 2x^2 + 4x^2 = (3 + 2 + 4)x^2 = 9x^2$$

Q: What is the difference between combining like terms and adding like terms?

A: Combining like terms involves adding the coefficients of like terms, while adding like terms involves adding the terms themselves. For example, in the polynomial $3x^2 + 2x^2 + 4x^2$, we can combine the like terms as follows:

$$3x^2 + 2x^2 + 4x^2 = (3 + 2 + 4)x^2 = 9x^2$$

However, if we add the terms themselves, we get:

$$3x^2 + 2x^2 + 4x^2 = 9x^2$$

Q: Can I simplify a polynomial by grouping like terms?

A: Yes, you can simplify a polynomial by grouping like terms. Grouping like terms can help to make the polynomial more manageable and easier to work with.

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

A: To check if you have grouped like terms correctly, make sure that the terms you are combining have the same variable(s) raised to the same power. If the terms have different variables or powers, they are not like terms and should not be combined.

Q: Can I use a calculator to simplify a polynomial by grouping like terms?

A: Yes, you can use a calculator to simplify a polynomial by grouping like terms. However, it's always a good idea to double-check your work by hand to make sure that you have grouped like terms correctly.

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

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

• Combining terms that are not like terms
• Forgetting to combine like terms
• Making errors when adding or subtracting coefficients

Q: How do I apply the concept of grouping like terms to real-world problems?

A: The concept of grouping like terms can be applied to a variety of real-world problems, such as:

• Simplifying complex equations in physics and engineering
• Analyzing data in statistics and data analysis
• Solving optimization problems in business and economics

Conclusion

Grouping like terms is an essential concept in algebra that can help to simplify complex polynomials and make them more manageable. By understanding how to group like terms, you can apply this concept to a variety of real-world problems and become a more confident and proficient mathematician.

Final Answer

The final answer is $\boxed{B}$.