Algebra I IC Sem B: Multiplying Polynomials And Simplifying ExpressionsSimplify The Expression $3x(x - 12x) + 3x^2 - 2(x - 2)^2$. Which Statements Are True About The Process And Simplified Product? Choose Three Correct Answers.A. The Term

Introduction

In Algebra I, students learn various techniques to simplify and manipulate algebraic expressions. One of the essential topics in this subject is multiplying polynomials and simplifying expressions. This article will focus on the process of simplifying expressions, specifically the given expression $3x(x - 12x) + 3x^2 - 2(x - 2)^2$. We will break down the expression, simplify it step by step, and identify the true statements about the process and simplified product.

Understanding the Expression

The given expression is a combination of two terms:

1. $$3x(x - 12x)$$

2. $$3x^2 - 2(x - 2)^2$$

To simplify the expression, we need to apply the distributive property and combine like terms.

Step 1: Simplify the First Term

The first term is $3x(x - 12x)$. To simplify this term, we need to apply the distributive property, which states that for any real numbers a, b, and c:

a(b + c) = ab + ac

Using this property, we can rewrite the first term as:

$$3x(x - 12x) = 3x \cdot x - 3x \cdot 12x$$

$$= 3x^2 - 36x^2$$

Step 2: Simplify the Second Term

The second term is $3x^2 - 2(x - 2)^2$. To simplify this term, we need to expand the squared term using the formula:

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

Applying this formula, we get:

$$-2(x - 2)^2 = -2(x^2 - 2 \cdot x \cdot 2 + 2^2)$$

$$= -2x^2 + 8x - 8$$

Step 3: Combine Like Terms

Now that we have simplified both terms, we can combine like terms to get the final simplified expression.

$$3x^2 - 36x^2 - 2x^2 + 8x - 8$$

$$= -35x^2 + 8x - 8$$

True Statements About the Process and Simplified Product

Based on the simplified expression, we can identify the following true statements:

1. The term -35x^2 is the result of combining like terms. When we combined the like terms -36x^2 and -2x^2, we got -35x^2.
2. The term 8x is the result of simplifying the second term. When we expanded the squared term and simplified it, we got 8x.
3. The term -8 is the result of simplifying the second term. When we expanded the squared term and simplified it, we got -8.

Conclusion

In this article, we simplified the expression $3x(x - 12x) + 3x^2 - 2(x - 2)^2$ step by step. We applied the distributive property, expanded the squared term, and combined like terms to get the final simplified expression. We also identified the true statements about the process and simplified product. By following these steps, students can simplify complex algebraic expressions and develop a deeper understanding of the subject.

Key Takeaways

• To simplify an expression, we need to apply the distributive property and combine like terms.
• The distributive property states that for any real numbers a, b, and c: a(b + c) = ab + ac.
• To expand a squared term, we can use the formula: (a - b)^2 = a^2 - 2ab + b^2.
• Combining like terms involves adding or subtracting the coefficients of the same variables.

Practice Problems

1. Simplify the expression: $2x(x + 3x) + 2x^2 - 3(x - 2)^2$
2. Simplify the expression: $4x(x - 2x) + 4x^2 - 2(x + 3)^2$
3. Simplify the expression: $3x(x + 2x) + 3x^2 - 2(x - 4)^2$

Answer Key

1. $$-x^2 + 6x + 2x^2 - 12$$

2. $$-8x^2 + 12x + 4x^2 - 18$$

3. -x^2 + 6x + 3x^2 - 16$<br/>

Introduction

In our previous article, we simplified the expression $3x(x - 12x) + 3x^2 - 2(x - 2)^2$ step by step. We applied the distributive property, expanded the squared term, and combined like terms to get the final simplified expression. In this article, we will answer some frequently asked questions related to multiplying polynomials and simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property allows us to distribute a single term to multiple terms inside the parentheses.

Q: How do I expand a squared term?

A: To expand a squared term, we can use the formula:

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

For example, to expand the term (x - 2)^2, we can use the formula:

(x - 2)^2 = x^2 - 2x \cdot 2 + 2^2

= x^2 - 4x + 4

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves adding or subtracting the coefficients of the same variables. Simplifying an expression, on the other hand, involves applying the distributive property, expanding squared terms, and combining like terms to get the final simplified expression.

Q: How do I know which terms to combine?

A: To combine like terms, we need to identify the terms with the same variables and coefficients. For example, in the expression 2x + 3x, we can combine the terms 2x and 3x to get 5x.

Q: Can I simplify an expression by just combining like terms?

A: No, simplifying an expression involves more than just combining like terms. We need to apply the distributive property, expand squared terms, and combine like terms to get the final simplified expression.

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

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

• Not applying the distributive property
• Not expanding squared terms
• Not combining like terms
• Not checking for errors in the final simplified expression

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working on practice problems, such as the ones listed below:

1. Simplify the expression: $2x(x + 3x) + 2x^2 - 3(x - 2)^2$
2. Simplify the expression: $4x(x - 2x) + 4x^2 - 2(x + 3)^2$
3. Simplify the expression: $3x(x + 2x) + 3x^2 - 2(x - 4)^2$

Answer Key

1. $$-x^2 + 6x + 2x^2 - 12$$

2. $$-8x^2 + 12x + 4x^2 - 18$$

3. $$-x^2 + 6x + 3x^2 - 16$$

Conclusion

In this article, we answered some frequently asked questions related to multiplying polynomials and simplifying expressions. We covered topics such as the distributive property, expanding squared terms, combining like terms, and common mistakes to avoid. By practicing simplifying expressions and following these tips, you can become more confident and proficient in simplifying complex algebraic expressions.

Key Takeaways

• The distributive property states that for any real numbers a, b, and c: a(b + c) = ab + ac.
• To expand a squared term, we can use the formula: (a - b)^2 = a^2 - 2ab + b^2.
• Combining like terms involves adding or subtracting the coefficients of the same variables.
• Simplifying an expression involves applying the distributive property, expanding squared terms, and combining like terms to get the final simplified expression.
• Common mistakes to avoid when simplifying expressions include not applying the distributive property, not expanding squared terms, not combining like terms, and not checking for errors in the final simplified expression.