The Expression { (4c - 3d)(3c + D)$}$ Is Equivalent To:A. ${ 12c^2 - 13cd - 3d^2\$} B. ${ 12c^2 - 13cd + 3d^2\$} C. ${ 12c^2 - 5cd - 3d^2\$} D. ${ 12c^2 - 5cd + 3d^2\$} E. ${ 12c^2 - 3d^2\$}
Introduction
In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. One common technique for simplifying expressions is the distributive property, which allows us to expand the product of two or more terms. In this article, we will use the distributive property to simplify the expression {(4c - 3d)(3c + d)$}$ and determine its equivalent form.
The Distributive Property
The distributive property states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property can be extended to the product of three or more terms. To apply the distributive property, we multiply each term in the first expression by each term in the second expression.
Simplifying the Expression
Let's apply the distributive property to the expression {(4c - 3d)(3c + d)$}$:
{(4c - 3d)(3c + d)$ = (4c)(3c) + (4c)(d) - (3d)(3c) - (3d)(d)$
Using the distributive property, we can simplify each term:
[(4c)(d) = 4cd$ [-(3d)(d) = -3d^2$
Now, let's combine like terms:
[$12c^2 + 4cd - 9cd - 3d^2$
Combining the like terms, we get:
[$12c^2 - 5cd - 3d^2$
Conclusion
In this article, we used the distributive property to simplify the expression [(4c - 3d)(3c + d)\$}. By applying the distributive property and combining like terms, we arrived at the simplified expression ${12c^2 - 5cd - 3d^2\$}. This expression is equivalent to the original expression, and it can be used to solve equations and manipulate mathematical statements.
Answer
The correct answer is:
C. ${12c^2 - 5cd - 3d^2\$}
Additional Examples
To further illustrate the concept of simplifying expressions using the distributive property, let's consider a few additional examples:
- {(2x + 3)(x - 4)$ = [2x^2 - 8x + 3x - 12}$ = ${2x^2 - 5x - 12}$
- {(x - 2)(x + 5)$ = [x^2 + 5x - 2x - 10}$ = {x^2 + 3x - 10}$
These examples demonstrate how the distributive property can be used to simplify expressions and arrive at equivalent forms.
Tips and Tricks
When simplifying expressions using the distributive property, remember to:
- Apply the distributive property to each term in the first expression
- Multiply each term in the first expression by each term in the second expression
- Combine like terms to simplify the expression
- Check your work by plugging in values or using a calculator to verify the result
By following these tips and tricks, you can master the art of simplifying expressions using the distributive property and become a proficient algebraist.
Conclusion
Q: What is the distributive property?
A: The distributive property is a mathematical concept that allows us to expand the product of two or more terms. It states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property can be extended to the product of three or more terms.
Q: How do I apply the distributive property?
A: To apply the distributive property, you need to multiply each term in the first expression by each term in the second expression. For example, if you have the expression (a + b)(c + d), you would multiply a by c, a by d, b by c, and b by d.
Q: What is the difference between the distributive property and the commutative property?
A: The distributive property and the commutative property are two separate mathematical concepts. The distributive property allows us to expand the product of two or more terms, while the commutative property states that the order of the terms does not change the result. For example, (a + b) + c = a + (b + c) is an example of the commutative property.
Q: Can I use the distributive property to simplify expressions with variables?
A: Yes, you can use the distributive property to simplify expressions with variables. For example, if you have the expression (2x + 3)(x - 4), you can apply the distributive property to expand the product and simplify the expression.
Q: How do I know when to use the distributive property?
A: You should use the distributive property when you have an expression that involves the product of two or more terms. The distributive property is particularly useful when you need to expand the product of two or more binomials.
Q: Can I use the distributive property to simplify expressions with fractions?
A: Yes, you can use the distributive property to simplify expressions with fractions. For example, if you have the expression (1/2x + 3/4)(x - 2), you can apply the distributive property to expand the product and simplify the expression.
Q: What are some common mistakes to avoid when using the distributive property?
A: Some common mistakes to avoid when using the distributive property include:
- Forgetting to multiply each term in the first expression by each term in the second expression
- Not combining like terms after applying the distributive property
- Making errors when multiplying variables or fractions
Q: How can I practice using the distributive property?
A: You can practice using the distributive property by working through examples and exercises in your algebra textbook or online resources. You can also try applying the distributive property to real-world problems or word problems to see how it can be used in different contexts.
Q: What are some real-world applications of the distributive property?
A: The distributive property has many real-world applications, including:
- Algebraic geometry: The distributive property is used to simplify expressions and manipulate mathematical statements in algebraic geometry.
- Computer science: The distributive property is used in computer science to simplify expressions and manipulate mathematical statements in programming languages.
- Engineering: The distributive property is used in engineering to simplify expressions and manipulate mathematical statements in the design and analysis of systems.
Conclusion
In conclusion, the distributive property is a powerful tool for simplifying expressions and manipulating mathematical statements. By understanding the distributive property and how to apply it, you can solve equations and manipulate mathematical statements with ease. Whether you're a student or a professional, mastering the distributive property is essential for success in algebra and beyond.