Find The Product. Simplify Your Answer. { (4u - 4)(2u + 3)$}$ { \square$}$

Mar 13, 2025 by ADMIN 75 views

Simplifying the Product: A Step-by-Step Guide

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying products of expressions. In this article, we will focus on simplifying the product of two expressions: $(4u - 4)(2u + 3)$ . We will break down the process into manageable steps, making it easier to understand and apply.

Understanding the Expression

The given expression is a product of two binomials: $(4u - 4)$ and $(2u + 3)$ . To simplify this expression, we will use the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ .

Step 1: Apply the Distributive Property

To simplify the expression, we will start by applying the distributive property to the first term, $4u$ . We will multiply $4u$ by each term in the second binomial, $(2u + 3)$ .

(4u - 4)(2u + 3) = 4u(2u + 3) - 4(2u + 3)

Step 2: Simplify the First Term

Now, we will simplify the first term, $4u(2u + 3)$ . We will use the distributive property again to multiply $4u$ by each term in the second binomial.

4u(2u + 3) = 4u(2u) + 4u(3)

Step 3: Simplify the Second Term

Next, we will simplify the second term, $-4(2u + 3)$ . We will use the distributive property again to multiply $-4$ by each term in the second binomial.

-4(2u + 3) = -4(2u) - 4(3)

Step 4: Combine Like Terms

Now, we will combine like terms in the expression. We will add or subtract the coefficients of like terms.

(4u - 4)(2u + 3) = 8u^2 + 12u - 8u - 12

Step 5: Simplify the Expression

Finally, we will simplify the expression by combining like terms.

(4u - 4)(2u + 3) = 8u^2 + 4u - 12

Conclusion

Simplifying the product of two expressions involves applying the distributive property and combining like terms. By following the steps outlined in this article, you can simplify the product of two binomials, $(4u - 4)(2u + 3)$ . Remember to apply the distributive property and combine like terms to simplify the expression.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ .

Q: How do I simplify a product of two expressions?

A: To simplify a product of two expressions, you will apply the distributive property and combine like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and coefficient.

Example Problems

Problem 1:

Simplify the product of the following expressions: $(3x + 2)(2x - 5)$ .

Solution:

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

(3x + 2)(2x - 5) = 6x^2 - 15x + 4x - 10

(3x + 2)(2x - 5) = 6x^2 - 11x - 10

Problem 2:

Simplify the product of the following expressions: $(4y - 3)(3y + 2)$ .

Solution:

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

(4y - 3)(3y + 2) = 12y^2 + 8y - 9y - 6

(4y - 3)(3y + 2) = 12y^2 - y - 6

Final Thoughts

Simplifying the product of two expressions is an essential skill in mathematics. By applying the distributive property and combining like terms, you can simplify complex expressions and solve problems more efficiently. Remember to practice regularly to develop your skills and become more confident in your ability to simplify products of expressions.

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Introduction

Simplifying the product of two expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying products of expressions. In this article, we will focus on providing answers to frequently asked questions about simplifying the product of two expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . This property allows us to multiply a single term by each term in a binomial.

Q: How do I simplify a product of two expressions?

A: To simplify a product of two expressions, you will apply the distributive property and combine like terms. This involves multiplying each term in the first expression by each term in the second expression and then combining like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and coefficient. For example, $2x$ and $4x$ are like terms because they both have the variable $x$ and the coefficient $2$ and $4$ respectively.

Q: How do I identify like terms?

A: To identify like terms, you will look for terms that have the same variable and coefficient. For example, in the expression $3x + 2x + 4y$ , the terms $3x$ and $2x$ are like terms because they both have the variable $x$ and the coefficient $3$ and $2$ respectively.

Q: Can I simplify a product of two expressions with variables and constants?

A: Yes, you can simplify a product of two expressions with variables and constants. To do this, you will apply the distributive property and combine like terms.

Q: How do I simplify a product of two expressions with exponents?

A: To simplify a product of two expressions with exponents, you will apply the distributive property and combine like terms. You will also need to use the rules of exponents to simplify the expression.

Q: Can I simplify a product of two expressions with negative coefficients?

A: Yes, you can simplify a product of two expressions with negative coefficients. To do this, you will apply the distributive property and combine like terms.

Q: How do I simplify a product of two expressions with fractions?

A: To simplify a product of two expressions with fractions, you will apply the distributive property and combine like terms. You will also need to use the rules of fractions to simplify the expression.

Example Problems

Problem 1:

Simplify the product of the following expressions: $(3x + 2)(2x - 5)$ .

Solution:

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

(3x + 2)(2x - 5) = 6x^2 - 15x + 4x - 10

(3x + 2)(2x - 5) = 6x^2 - 11x - 10

Problem 2:

Simplify the product of the following expressions: $(4y - 3)(3y + 2)$ .

Solution:

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

(4y - 3)(3y + 2) = 12y^2 + 8y - 9y - 6

(4y - 3)(3y + 2) = 12y^2 - y - 6

Final Thoughts

Simplifying the product of two expressions is an essential skill in mathematics. By understanding the distributive property and combining like terms, you can simplify complex expressions and solve problems more efficiently. Remember to practice regularly to develop your skills and become more confident in your ability to simplify products of expressions.

Additional Resources

Online Resources

Khan Academy: Simplifying Expressions
Mathway: Simplifying Expressions
Wolfram Alpha: Simplifying Expressions

Textbooks

"Algebra" by Michael Artin
"Calculus" by Michael Spivak
"Mathematics for Computer Science" by Eric Lehman

Practice Problems

Simplify the product of the following expressions: $(2x + 3)(x - 4)$ .
Simplify the product of the following expressions: $(4y - 2)(3y + 5)$ .
Simplify the product of the following expressions: $(x + 2)(x - 3)$ .

Conclusion

Simplifying the product of two expressions is a crucial skill in mathematics. By understanding the distributive property and combining like terms, you can simplify complex expressions and solve problems more efficiently. Remember to practice regularly to develop your skills and become more confident in your ability to simplify products of expressions.