Which Is A Correct Expansion Of $(3x + 2)(3x^2 + 4)$?A. $3x \cdot 3x^2 + 2 \cdot 3x^2 + 3x \cdot 4 + 2 \cdot 4$B. $ 3 X ⋅ 3 X 2 + 3 X ⋅ 4 + 2 ⋅ 3 X + 2 ⋅ 4 3x \cdot 3x^2 + 3x \cdot 4 + 2 \cdot 3x + 2 \cdot 4 3 X ⋅ 3 X 2 + 3 X ⋅ 4 + 2 ⋅ 3 X + 2 ⋅ 4 [/tex]C. $3x \cdot 3x^2 + 3x \cdot 4 + 2

Introduction

Algebraic expressions are a fundamental concept in mathematics, and expanding them is a crucial skill to master. In this article, we will explore the correct expansion of the given expression $(3x + 2)(3x^2 + 4)$ and provide a step-by-step guide on how to approach similar problems.

Understanding the Expression

The given expression is a product of two binomials, $(3x + 2)$ and $(3x^2 + 4)$. To expand this expression, we need to apply 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 expand the given expression, we need to multiply each term in the first binomial by each term in the second binomial. This can be done by applying the distributive property twice.

(3x + 2)(3x^2 + 4) = (3x)(3x^2) + (3x)(4) + (2)(3x^2) + (2)(4)

Step 2: Simplify the Expression

Now that we have applied the distributive property, we can simplify the expression by combining like terms.

(3x)(3x^2) = 9x^3
(3x)(4) = 12x
(2)(3x^2) = 6x^2
(2)(4) = 8

Step 3: Combine Like Terms

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

9x^3 + 12x + 6x^2 + 8

Conclusion

In conclusion, the correct expansion of the given expression $(3x + 2)(3x^2 + 4)$ is $9x^3 + 12x + 6x^2 + 8$. This can be achieved by applying the distributive property and simplifying the expression by combining like terms.

Common Mistakes to Avoid

When expanding algebraic expressions, it's essential to avoid common mistakes. Here are a few tips to help you avoid mistakes:

• Make sure to apply the distributive property correctly.
• Simplify the expression by combining like terms.
• Check your work by plugging in values or using a calculator.

Real-World Applications

Expanding algebraic expressions has numerous real-world applications. Here are a few examples:

• Physics: Algebraic expressions are used to describe the motion of objects in physics. For example, the equation of motion for an object under constant acceleration is given by $s = ut + \frac{1}{2}at^2$, where $s$ is the displacement, $u$ is the initial velocity, $t$ is time, and $a$ is the acceleration.
• Engineering: Algebraic expressions are used to design and optimize systems in engineering. For example, the equation for the stress on a beam is given by $\sigma = \frac{F}{A}$, where $\sigma$ is the stress, $F$ is the force, and $A$ is the cross-sectional area.
• Computer Science: Algebraic expressions are used to represent and manipulate data in computer science. For example, the equation for the cost of a product is given by $C = P \times Q$, where $C$ is the cost, $P$ is the price, and $Q$ is the quantity.

Conclusion

In conclusion, expanding algebraic expressions is a crucial skill in mathematics and has numerous real-world applications. By following the steps outlined in this article, you can master the art of expanding algebraic expressions and apply it to solve problems in physics, engineering, and computer science.

Final Answer

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This means that we can multiply a single term by two or more terms inside a set of parentheses.

Q: How do I apply the distributive property to expand an algebraic expression?

A: To apply the distributive property, we need to multiply each term in the first binomial by each term in the second binomial. This can be done by following these steps:

1. Multiply each term in the first binomial by each term in the second binomial.
2. Simplify the expression by combining like terms.
3. Check your work by plugging in values or using a calculator.

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

A: Expanding an algebraic expression involves applying the distributive property to multiply two or more binomials. Simplifying an algebraic expression involves combining like terms to reduce the expression to its simplest form.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, we need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent. For example, $2x + 3x$ can be simplified to $5x$ by combining the like terms.

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

A: Some common mistakes to avoid when expanding and simplifying algebraic expressions include:

• Failing to apply the distributive property correctly.
• Failing to simplify the expression by combining like terms.
• Making errors when multiplying or adding terms.

Q: How do I check my work when expanding and simplifying algebraic expressions?

A: To check your work, you can plug in values or use a calculator to evaluate the expression. You can also use a graphing calculator or a computer algebra system to check your work.

Q: What are some real-world applications of expanding and simplifying algebraic expressions?

A: Expanding and simplifying algebraic expressions has numerous real-world applications, including:

• Physics: Algebraic expressions are used to describe the motion of objects in physics.
• Engineering: Algebraic expressions are used to design and optimize systems in engineering.
• Computer Science: Algebraic expressions are used to represent and manipulate data in computer science.

Q: How do I practice expanding and simplifying algebraic expressions?

A: To practice expanding and simplifying algebraic expressions, you can try the following:

• Start with simple expressions and gradually move on to more complex ones.
• Use online resources or worksheets to practice expanding and simplifying algebraic expressions.
• Work with a partner or tutor to get feedback and guidance.

Conclusion

In conclusion, expanding and simplifying algebraic expressions is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined in this article and practicing regularly, you can master the art of expanding and simplifying algebraic expressions and apply it to solve problems in physics, engineering, and computer science.

Final Answer

The final answer is: $\boxed{9x^3 + 12x + 6x^2 + 8}$