Which Is Equivalent To { (4xy - 3z)^2$}$, And What Type Of Special Product Is It?A. ${ 16x^2y^2 + 9z^2\$} , The Difference Of Squares B. ${ 16x^2y^2 + 9z^2\$} , A Perfect Square Trinomial C. [$16x 2y 2 - 24xyz +
Introduction
Algebraic expressions are a fundamental part of mathematics, and understanding how to expand and simplify them is crucial for solving equations and inequalities. One of the most important concepts in algebra is the expansion of special products, which can be used to simplify complex expressions. In this article, we will explore the expansion of the expression {(4xy - 3z)^2$}$ and determine what type of special product it is.
The Concept of Special Products
Special products are algebraic expressions that can be expanded using specific formulas. These formulas allow us to simplify complex expressions and make them easier to work with. There are several types of special products, including the difference of squares, the sum of cubes, and the perfect square trinomial.
The Difference of Squares
The difference of squares is a special product that can be expanded using the formula {a^2 - b^2 = (a + b)(a - b)$}$. This formula can be used to simplify expressions that involve the subtraction of two squared terms.
The Perfect Square Trinomial
The perfect square trinomial is a special product that can be expanded using the formula {a^2 + 2ab + b^2 = (a + b)^2$}$ or {a^2 - 2ab + b^2 = (a - b)^2$}$. This formula can be used to simplify expressions that involve the addition or subtraction of two squared terms and a middle term.
Expanding the Expression {(4xy - 3z)^2$}$
To expand the expression {(4xy - 3z)^2$}$, we can use the formula for the difference of squares. This formula states that {a^2 - b^2 = (a + b)(a - b)$}$. In this case, we can let {a = 4xy$}$ and {b = 3z$}$.
Using the formula, we can expand the expression as follows:
{(4xy - 3z)^2 = (4xy + 3z)(4xy - 3z)$}$
Expanding the expression further, we get:
{(4xy + 3z)(4xy - 3z) = 16x2y2 - 9z^2$}$
Conclusion
In conclusion, the expression {(4xy - 3z)^2$}$ is equivalent to ${16x^2y^2 - 9z^2\$}, which is a difference of squares. This type of special product can be expanded using the formula {a^2 - b^2 = (a + b)(a - b)$}$. Understanding how to expand and simplify special products is crucial for solving equations and inequalities, and is an important concept in algebra.
What is the Type of Special Product?
The expression {(4xy - 3z)^2$}$ is a difference of squares, which is a type of special product. This type of special product can be expanded using the formula {a^2 - b^2 = (a + b)(a - b)$}$.
Why is it Important to Understand Special Products?
Understanding special products is important because it allows us to simplify complex expressions and make them easier to work with. Special products can be used to solve equations and inequalities, and are an important concept in algebra.
Real-World Applications of Special Products
Special products have many real-world applications, including:
- Physics: Special products are used to describe the motion of objects in physics.
- Engineering: Special products are used to design and build complex systems in engineering.
- Computer Science: Special products are used to develop algorithms and data structures in computer science.
Conclusion
In conclusion, the expression {(4xy - 3z)^2$}$ is equivalent to ${16x^2y^2 - 9z^2\$}, which is a difference of squares. This type of special product can be expanded using the formula {a^2 - b^2 = (a + b)(a - b)$}$. Understanding how to expand and simplify special products is crucial for solving equations and inequalities, and is an important concept in algebra.
Final Answer
The final answer is:
- **A. $16x^2y^2 + 9z^2\$}, the difference of squares**$ is equivalent to ${16x^2y^2 - 9z^2\$}, not ${16x^2y^2 + 9z^2\$}.
- **B. $16x^2y^2 + 9z^2\$}, a perfect square trinomial**$ is equivalent to ${16x^2y^2 - 9z^2\$}, not ${16x^2y^2 + 9z^2\$}.
- **C. $16x^2y^2 - 24xyz + 9z^2\$}, the perfect square trinomial**$ is equivalent to ${16x^2y^2 - 9z^2\$}, not ${16x^2y^2 - 24xyz + 9z^2\$}.
The correct answer is:
- **A. $16x^2y^2 - 9z^2\$}, the difference of squares**$ is equivalent to ${16x^2y^2 - 9z^2\$}, which is a difference of squares.
Q&A: Expanding and Simplifying Algebraic Expressions =====================================================
Q: What is the difference of squares?
A: The difference of squares is a special product that can be expanded using the formula {a^2 - b^2 = (a + b)(a - b)$}$. This formula can be used to simplify expressions that involve the subtraction of two squared terms.
Q: How do I expand the expression {(4xy - 3z)^2$}$?
A: To expand the expression {(4xy - 3z)^2$}$, you can use the formula for the difference of squares. This formula states that {a^2 - b^2 = (a + b)(a - b)$}$. In this case, you can let {a = 4xy$}$ and {b = 3z$}$.
Using the formula, you can expand the expression as follows:
{(4xy - 3z)^2 = (4xy + 3z)(4xy - 3z)$}$
Expanding the expression further, you get:
{(4xy + 3z)(4xy - 3z) = 16x2y2 - 9z^2$}$
Q: What type of special product is the expression {(4xy - 3z)^2$}$?
A: The expression {(4xy - 3z)^2$}$ is a difference of squares, which is a type of special product. This type of special product can be expanded using the formula {a^2 - b^2 = (a + b)(a - b)$}$.
Q: Why is it important to understand special products?
A: Understanding special products is important because it allows us to simplify complex expressions and make them easier to work with. Special products can be used to solve equations and inequalities, and are an important concept in algebra.
Q: What are some real-world applications of special products?
A: Special products have many real-world applications, including:
- Physics: Special products are used to describe the motion of objects in physics.
- Engineering: Special products are used to design and build complex systems in engineering.
- Computer Science: Special products are used to develop algorithms and data structures in computer science.
Q: How do I determine if an expression is a difference of squares?
A: To determine if an expression is a difference of squares, you can look for the following pattern:
{a^2 - b^2$}$
If the expression matches this pattern, then it is a difference of squares.
Q: Can you give me some examples of difference of squares?
A: Yes, here are some examples of difference of squares:
- ${16x^2 - 9y^2 = (4x + 3y)(4x - 3y)\$}
- ${25a^2 - 36b^2 = (5a + 6b)(5a - 6b)\$}
- ${49c^2 - 16d^2 = (7c + 4d)(7c - 4d)\$}
Q: How do I expand a difference of squares?
A: To expand a difference of squares, you can use the formula {a^2 - b^2 = (a + b)(a - b)$}$. This formula can be used to simplify expressions that involve the subtraction of two squared terms.
Q: Can you give me some tips for working with special products?
A: Yes, here are some tips for working with special products:
- Pay attention to the signs: When working with special products, pay attention to the signs of the terms. This will help you to determine if the expression is a difference of squares or a perfect square trinomial.
- Use the formulas: Use the formulas for special products to simplify expressions. This will help you to avoid making mistakes and to get the correct answer.
- Practice, practice, practice: The more you practice working with special products, the more comfortable you will become with the formulas and the easier it will be to simplify expressions.
Conclusion
In conclusion, understanding special products is an important concept in algebra. Special products can be used to simplify complex expressions and make them easier to work with. By following the tips and examples in this article, you can become more comfortable working with special products and simplify expressions with ease.