Select The Correct Answer.What Is The Simplest Form Of This Expression? $ (2x - 3)(3x^2 + 2x - 1) $A. $ 6x^3 - 5x^2 - 6x + 2 $ B. $ 6x^3 - 5x^2 - 8x + 3 $ C. $ 6x^3 - 9x^2 - 4x + 3 $ D. $ 6x^3 - 2x^2 - 8x + 3

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 expression $(2x - 3)(3x^2 + 2x - 1)$ as an example and guide you through the step-by-step process of simplifying it.

Understanding the Expression

Before we dive into the simplification process, let's take a closer look at the given expression:

$$(2x - 3)(3x^2 + 2x - 1)$$

This expression is a product of two binomials, which means we can use the distributive property to expand it. The distributive property states that for any real numbers $a$, $b$, and $c$:

$$a(b + c) = ab + ac$$

Step 1: Apply the Distributive Property

To simplify the given expression, we will apply the distributive property to each term in the first binomial $(2x - 3)$ and multiply it with each term in the second binomial $(3x^2 + 2x - 1)$.

(2x - 3)(3x^2 + 2x - 1)
= (2x)(3x^2) + (2x)(2x) + (2x)(-1) + (-3)(3x^2) + (-3)(2x) + (-3)(-1)

Step 2: Simplify Each Term

Now that we have applied the distributive property, let's simplify each term:

= 6x^3 + 4x^2 - 2x - 9x^2 - 6x + 3

Step 3: Combine Like Terms

The next step is to combine like terms, which means combining terms with the same variable and exponent. In this case, we have two terms with the variable $x^2$ and two terms with the variable $x$.

= 6x^3 + (-9x^2 + 4x^2) + (-2x - 6x) + 3
= 6x^3 - 5x^2 - 8x + 3

Conclusion

In conclusion, the simplest form of the given expression $(2x - 3)(3x^2 + 2x - 1)$ is $6x^3 - 5x^2 - 8x + 3$. This was achieved by applying the distributive property, simplifying each term, and combining like terms.

Answer

The correct answer is:

• B. $6x^3 - 5x^2 - 8x + 3$

Discussion

This problem requires a good understanding of the distributive property and how to apply it to simplify algebraic expressions. It also requires the ability to combine like terms and simplify the resulting expression. If you have any questions or need further clarification, please don't hesitate to ask.

Additional Resources

If you want to learn more about simplifying algebraic expressions or need additional practice, here are some resources you can use:

• Khan Academy: Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• IXL: Algebraic Expressions

Final Thoughts

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

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$:

$$a(b + c) = ab + ac$$

This property is used to simplify algebraic expressions by distributing each term in the first binomial to each term in the second binomial.

Q: How do I apply the distributive property to simplify an algebraic expression?

A: To apply the distributive property, follow these steps:

1. Identify the two binomials in the expression.
2. Multiply each term in the first binomial to each term in the second binomial.
3. Simplify each term by combining like terms.
4. Combine like terms to get the simplest form of the expression.

Q: What are like terms, and how do I combine them?

A: Like terms are terms that have the same variable and exponent. To combine like terms, add or subtract the coefficients of the terms.

For example, if you have the expression $2x^2 + 3x^2$, you can combine the like terms by adding the coefficients:

$$2x^2 + 3x^2 = (2 + 3)x^2 = 5x^2$$

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, follow these steps:

1. Identify the variables and their exponents in the expression.
2. Apply the distributive property to each term in the expression.
3. Simplify each term by combining like terms.
4. Combine like terms to get the simplest form of the expression.

For example, if you have the expression $(2x + 3y)(x^2 + 2y)$, you can simplify it by applying the distributive property:

$$(2x + 3y)(x^2 + 2y) = 2x(x^2 + 2y) + 3y(x^2 + 2y)$$

Simplifying each term, you get:

$$2x^3 + 4xy + 3x^2y + 6y^2$$

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

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

• Forgetting to apply the distributive property
• Not combining like terms
• Making errors when simplifying each term
• Not checking the final expression for errors

Q: How can I practice simplifying algebraic expressions?

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

• Online algebraic expression simplifiers, such as Mathway or IXL
• Algebra textbooks or workbooks
• Online algebra courses or tutorials
• Practice problems or worksheets

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 in physics or engineering
• Modeling population growth or decline in biology
• Analyzing data in statistics or economics
• Solving optimization problems in business or finance

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it requires practice and patience. By following the steps outlined in this article and practicing with real-world examples, you can become proficient in simplifying algebraic expressions and apply them to a variety of fields.