Introduction to Algebraic Expansion

Oct 13, 2025 by talynt.org 121 views

Introduction to Algebraic Expansion

In algebra, expansion is a process of multiplying and combining like terms to simplify an expression. The given expression, $$4(4n + 3q - r)$$, is a quadratic expression that needs to be expanded using the distributive property. This property states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses. In this case, we have a term 4 outside the parentheses and a quadratic expression 4n + 3q - r inside the parentheses.

Step-by-Step Expansion

To expand the given expression, we will apply the distributive property by multiplying each term inside the parentheses with the term outside the parentheses.

Multiply 4 with 4n

The first step is to multiply 4 with 4n. This can be written as:

4 × 4n = 16n

Multiply 4 with 3q

Next, we multiply 4 with 3q:

4 × 3q = 12q

Multiply 4 with -r

Finally, we multiply 4 with -r:

4 × -r = -4r

Combining Like Terms

Now that we have expanded the expression by multiplying each term inside the parentheses with the term outside the parentheses, we can combine like terms to simplify the expression.

Combine the Constant Terms

The constant terms in the expression are 16n, 12q, and -4r. Since these terms have different variables, we cannot combine them. However, we can rewrite the expression as:

16n + 12q - 4r

Final Answer

The expanded form of the given expression $$4(4n + 3q - r)$$ is:

16n + 12q - 4r

This is the final answer to the problem.

Conclusion

In this article, we expanded the given expression $$4(4n + 3q - r)$$ using the distributive property. We applied the distributive property by multiplying each term inside the parentheses with the term outside the parentheses and then combined like terms to simplify the expression. The final answer is 16n + 12q - 4r.

Frequently Asked Questions

Q: What is the distributive property in algebra?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses.

Q: How do I expand an expression using the distributive property?

A: To expand an expression using the distributive property, multiply each term inside the parentheses with the term outside the parentheses and then combine like terms.

Q: What are like terms in algebra?

A: Like terms in algebra are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the same exponent.

Introduction

In our previous article, we expanded the expression $$4(4n + 3q - r)$$ using the distributive property. In this article, we will answer some frequently asked questions about algebra expansion.

Q&A

Q: What is the distributive property in algebra?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses.

Q: How do I expand an expression using the distributive property?

A: To expand an expression using the distributive property, multiply each term inside the parentheses with the term outside the parentheses and then combine like terms.

Q: What are like terms in algebra?

A: Like terms in algebra are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the same exponent.

Q: How do I combine like terms in an expression?

A: To combine like terms in an expression, add or subtract the coefficients of the like terms. For example, 2x + 4x = 6x.

Q: What is the difference between expanding and simplifying an expression?

A: Expanding an expression involves multiplying each term inside the parentheses with the term outside the parentheses, while simplifying an expression involves combining like terms and removing any unnecessary parentheses.

Q: Can I expand an expression with multiple sets of parentheses?

A: Yes, you can expand an expression with multiple sets of parentheses by applying the distributive property multiple times.

Q: How do I expand an expression with variables and constants?

A: To expand an expression with variables and constants, multiply each term inside the parentheses with the term outside the parentheses, just like you would with variables.

Q: Can I use the distributive property to expand an expression with fractions?

A: Yes, you can use the distributive property to expand an expression with fractions. Just multiply each term inside the parentheses with the term outside the parentheses, just like you would with variables.

Q: How do I expand an expression with exponents?

A: To expand an expression with exponents, multiply each term inside the parentheses with the term outside the parentheses, just like you would with variables.

Q: Can I use the distributive property to expand an expression with negative numbers?

A: Yes, you can use the distributive property to expand an expression with negative numbers. Just multiply each term inside the parentheses with the term outside the parentheses, just like you would with variables.

Examples

Example 1: Expanding an Expression with Variables

Expand the expression $$3(2x + 4y - 2z)$$.

Solution:

Multiply 3 with 2x: 3 × 2x = 6x
Multiply 3 with 4y: 3 × 4y = 12y
Multiply 3 with -2z: 3 × -2z = -6z
Combine like terms: 6x + 12y - 6z

Example 2: Expanding an Expression with Fractions

Expand the expression $$\frac{1}{2}(3x + 2y - 4z)$$.

Solution:

Multiply 1/2 with 3x: 1/2 × 3x = 3/2x
Multiply 1/2 with 2y: 1/2 × 2y = y
Multiply 1/2 with -4z: 1/2 × -4z = -2z
Combine like terms: 3/2x + y - 2z

Example 3: Expanding an Expression with Exponents

Expand the expression $$2(3x^2 + 2y^3 - 4z^4)$$.

Solution:

Multiply 2 with 3x^2: 2 × 3x^2 = 6x^2
Multiply 2 with 2y^3: 2 × 2y^3 = 4y^3
Multiply 2 with -4z^4: 2 × -4z^4 = -8z^4
Combine like terms: 6x^2 + 4y^3 - 8z^4

Conclusion

In this article, we answered some frequently asked questions about algebra expansion. We also provided examples of expanding expressions with variables, fractions, and exponents. Remember to always apply the distributive property and combine like terms to simplify expressions.