Add: ( 16 B C − 12 C D ) + ( − 8 C D + 6 B C + 4 B D (16bc - 12cd) + (-8cd + 6bc + 4bd ( 16 B C − 12 C D ) + ( − 8 C D + 6 B C + 4 B D ]A. 22 B C − 20 C D + 4 B D 22bc - 20cd + 4bd 22 B C − 20 C D + 4 B D B. 22 B 2 C 2 − 20 C 2 D 2 + 4 B D 22b^2c^2 - 20c^2d^2 + 4bd 22 B 2 C 2 − 20 C 2 D 2 + 4 B D C. 6 B C D 6bcd 6 B C D D. 8 B C − 6 C D + 4 B D 8bc - 6cd + 4bd 8 B C − 6 C D + 4 B D

Mar 7, 2025 by ADMIN 385 views

**Simplifying Algebraic Expressions: A Step-by-Step Guide**

Understanding the Problem

In this article, we will focus on simplifying algebraic expressions, specifically the given expression: $(16bc - 12cd) + (-8cd + 6bc + 4bd)$ . We will break down the problem into manageable steps and provide a clear explanation of each step.

Step 1: Distribute the Negative Sign

The first step in simplifying the given expression is to distribute the negative sign to the terms inside the second set of parentheses. This will change the sign of each term inside the parentheses.

(-8cd + 6bc + 4bd) = -8cd + 6bc + 4bd

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable(s) raised to the same power.

(16bc - 12cd) + (-8cd + 6bc + 4bd) = 16bc - 12cd - 8cd + 6bc + 4bd

Step 3: Combine the Constant Terms

The next step is to combine the constant terms. In this case, we have two constant terms: $-12cd$ and $-8cd$ . We can combine these terms by adding their coefficients.

-12cd - 8cd = -20cd

Step 4: Combine the Variable Terms

Now that we have combined the constant terms, we can combine the variable terms. In this case, we have two variable terms: $16bc$ and $6bc$ . We can combine these terms by adding their coefficients.

16bc + 6bc = 22bc

Step 5: Write the Final Answer

Now that we have combined all the like terms, we can write the final answer.

(16bc - 12cd) + (-8cd + 6bc + 4bd) = 22bc - 20cd + 4bd

Conclusion

In this article, we have simplified the given algebraic expression by distributing the negative sign, combining like terms, and combining constant and variable terms. The final answer is $22bc - 20cd + 4bd$ .

Answer Key

The correct answer is A. $22bc - 20cd + 4bd$ .

Tips and Tricks

When simplifying algebraic expressions, it's essential to distribute the negative sign to the terms inside the parentheses.
Combine like terms by adding or subtracting their coefficients.
Combine constant and variable terms separately to avoid confusion.

Common Mistakes

Failing to distribute the negative sign to the terms inside the parentheses.
Not combining like terms correctly.
Combining constant and variable terms together.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in various fields, including mathematics, physics, engineering, and computer science. It's essential to understand how to simplify expressions to solve problems and make calculations more efficient.

Practice Problems

Simplify the expression: $(3x^2 - 2y^2) + (x^2 + 4y^2)$ .
Simplify the expression: $(2a^2 - 3b^2) + (-a^2 + 2b^2)$ .
Simplify the expression: $(4x^3 - 2y^3) + (x^3 + 3y^3)$ .

References

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow these steps:

Distribute the negative sign to the terms inside the parentheses.
Combine like terms by adding or subtracting their coefficients.
Combine constant and variable terms separately to avoid confusion.

Q: What is the difference between a constant and a variable?

A: A constant is a number that does not change value, while a variable is a letter or symbol that represents a value that can change.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, $2x^2 + 3x^2 = 5x^2$ .

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an algebraic expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression with parentheses?

A: To simplify an expression with parentheses, you need to follow these steps:

Distribute the negative sign to the terms inside the parentheses.
Combine like terms by adding or subtracting their coefficients.
Combine constant and variable terms separately to avoid confusion.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression that has a degree of 1, while a quadratic expression is an expression that has a degree of 2.

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you need to follow these steps:

Distribute the negative sign to the terms inside the parentheses.
Combine like terms by adding or subtracting their coefficients.
Combine constant and variable terms separately to avoid confusion.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

Failing to distribute the negative sign to the terms inside the parentheses.
Not combining like terms correctly.
Combining constant and variable terms together.

Q: How do I check my work when simplifying algebraic expressions?

A: To check your work when simplifying algebraic expressions, you need to follow these steps:

Simplify the expression using the order of operations.
Check that you have combined like terms correctly.
Check that you have combined constant and variable terms separately.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

Solving systems of equations
Finding the maximum or minimum value of a function
Modeling real-world phenomena

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

Working through practice problems
Using online resources and tools
Asking a teacher or tutor for help

Q: What are some resources for learning more about simplifying algebraic expressions?

A: Some resources for learning more about simplifying algebraic expressions include:

Online tutorials and videos
Textbooks and workbooks
Online communities and forums

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can simplify algebraic expressions with confidence. Remember to practice regularly and seek help when needed.