Select The Correct Answer.Simplify: $(x-7)(6x-3)$A. $7x^2 - 10$ B. $6x^2 + 39x - 21$ C. $ 6 X 2 + 21 6x^2 + 21 6 X 2 + 21 [/tex] D. $6x^2 - 45x + 21$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific type of algebraic expression, namely the product of two binomials. We will use the given problem as a case study to demonstrate the step-by-step process of simplifying algebraic expressions.

The Problem

The problem we will be working on is:

(xβˆ’7)(6xβˆ’3)(x-7)(6x-3)

We are asked to simplify this expression and choose the correct answer from the options provided.

Step 1: Distribute the First Term

To simplify the expression, we need to distribute the first term, (xβˆ’7)(x-7), to the second term, (6xβˆ’3)(6x-3). This involves multiplying each term in the first binomial by each term in the second binomial.

(xβˆ’7)(6xβˆ’3)=x(6xβˆ’3)βˆ’7(6xβˆ’3)(x-7)(6x-3) = x(6x-3) - 7(6x-3)

Step 2: Distribute the Second Term

Now, we need to distribute the second term, βˆ’7-7, to the second binomial, (6xβˆ’3)(6x-3). This involves multiplying each term in the second binomial by βˆ’7-7.

x(6xβˆ’3)βˆ’7(6xβˆ’3)=6x2βˆ’3xβˆ’42x+21x(6x-3) - 7(6x-3) = 6x^2 - 3x - 42x + 21

Step 3: Combine Like Terms

The next step is to combine like terms. In this case, we have two terms with the variable xx, namely βˆ’3x-3x and βˆ’42x-42x. We can combine these terms by adding their coefficients.

6x2βˆ’3xβˆ’42x+21=6x2βˆ’45x+216x^2 - 3x - 42x + 21 = 6x^2 - 45x + 21

Conclusion

In conclusion, the simplified expression is:

6x2βˆ’45x+216x^2 - 45x + 21

This expression is the result of distributing the first term to the second term and then combining like terms.

Choosing the Correct Answer

Now that we have simplified the expression, we can choose the correct answer from the options provided. The correct answer is:

  • D. 6x2βˆ’45x+216x^2 - 45x + 21

This is the only option that matches the simplified expression we obtained.

Why is this Important?

Simplifying algebraic expressions is an essential skill in mathematics, and it has numerous applications in various fields, such as physics, engineering, and economics. By simplifying expressions, we can:

  • Solve equations: Simplifying expressions is a crucial step in solving equations. By simplifying the expression, we can isolate the variable and solve for its value.
  • Analyze functions: Simplifying expressions is also important in analyzing functions. By simplifying the expression, we can identify the domain and range of the function.
  • Make predictions: Simplifying expressions can help us make predictions about the behavior of a system or a process. By simplifying the expression, we can identify the underlying patterns and relationships.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

  • Use the distributive property: The distributive property is a fundamental concept in algebra, and it can help you simplify expressions. By distributing the terms, you can break down the expression into smaller parts and simplify it.
  • Combine like terms: Combining like terms is an essential step in simplifying expressions. By combining like terms, you can eliminate unnecessary terms and simplify the expression.
  • Use algebraic identities: Algebraic identities are formulas that can help you simplify expressions. By using algebraic identities, you can simplify expressions and make them more manageable.

Conclusion

Introduction

In our previous article, we discussed the importance of simplifying algebraic expressions and provided a step-by-step guide on how to simplify a specific type of expression, namely the product of two binomials. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying algebraic expressions.

Q: What is the purpose of simplifying algebraic expressions?

A: The purpose of simplifying algebraic expressions is to make them more manageable and easier to work with. By simplifying expressions, you can:

  • Solve equations: Simplifying expressions is a crucial step in solving equations. By simplifying the expression, you can isolate the variable and solve for its value.
  • Analyze functions: Simplifying expressions is also important in analyzing functions. By simplifying the expression, you can identify the domain and range of the function.
  • Make predictions: Simplifying expressions can help you make predictions about the behavior of a system or a process. By simplifying the expression, you can identify the underlying patterns and relationships.

Q: What are the steps involved in simplifying algebraic expressions?

A: The steps involved in simplifying algebraic expressions are:

  1. Distribute the terms: Distribute the terms in the expression to break it down into smaller parts.
  2. Combine like terms: Combine like terms to eliminate unnecessary terms and simplify the expression.
  3. Use algebraic identities: Use algebraic identities to simplify expressions and make them more manageable.

Q: What is the distributive property, and how is it used in simplifying expressions?

A: The distributive property is a fundamental concept in algebra, and it can help you simplify expressions. By distributing the terms, you can break down the expression into smaller parts and simplify it. The distributive property states that:

  • a(b + c) = ab + ac

This means that you can distribute the term a to both terms b and c and simplify the expression.

Q: What are like terms, and how are they combined in simplifying expressions?

A: Like terms are terms that have the same variable and exponent. In simplifying expressions, like terms are combined by adding their coefficients. For example:

  • 2x + 3x = 5x

In this example, the terms 2x and 3x are like terms, and they are combined by adding their coefficients to get 5x.

Q: What are algebraic identities, and how are they used in simplifying expressions?

A: Algebraic identities are formulas that can help you simplify expressions. By using algebraic identities, you can simplify expressions and make them more manageable. Some common algebraic identities include:

  • a^2 + b^2 = (a + b)^2
  • a^2 - b^2 = (a + b)(a - b)

These identities can help you simplify expressions and make them more manageable.

Q: How can I practice simplifying algebraic expressions?

A: There are many ways to practice simplifying algebraic expressions, including:

  • Solving problems: Practice solving problems that involve simplifying algebraic expressions.
  • Using online resources: Use online resources, such as Khan Academy or Mathway, to practice simplifying algebraic expressions.
  • Working with a tutor: Work with a tutor or teacher to practice simplifying algebraic expressions.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill in mathematics, and it has numerous applications in various fields. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying algebraic expressions and tackle complex problems with confidence. Remember to use the distributive property, combine like terms, and use algebraic identities to simplify expressions. With practice and patience, you can become a master of simplifying algebraic expressions.