Equivalent Expressions For (3x^2 + 4x - 7)(x - 2) Explained

When dealing with algebraic expressions, it’s essential to understand how to manipulate them while preserving their value. This article dives deep into the expression $(3x^2 + 4x - 7)(x - 2)$, exploring which of the provided options is equivalent. We'll break down the process step by step, ensuring a clear understanding of the underlying principles.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra, allowing us to multiply a single term by multiple terms within parentheses. It states that $a(b + c) = ab + ac$. This property is crucial for expanding expressions and identifying equivalent forms.

In our case, we need to distribute $(3x^2 + 4x - 7)$ by $(x - 2)$. This means each term in the first expression must be multiplied by each term in the second expression. Let's analyze the given options in the context of this property.

Option A: (3x^2 + 4x - 7) + 2(3x^2 + 4x - 7)

This option suggests adding the original expression to twice itself. While this might seem like a valid operation, it's not equivalent to the original multiplication. This can be rewritten as $3(3x^2 + 4x - 7)$, which is a different operation than multiplying by $(x - 2)$. Therefore, option A is incorrect. To elaborate further, this option essentially triples the expression $(3x^2 + 4x - 7)$ rather than multiplying it by a binomial. This significantly alters the resulting polynomial, making it non-equivalent to the initial expression. The distributive property in this context would apply as $1*(3x^2 + 4x - 7) + 2*(3x^2 + 4x - 7)$, which simplifies to $3*(3x^2 + 4x - 7)$. This misunderstanding of the distributive property highlights a key difference between addition and multiplication in algebraic manipulations. Thus, while addition can combine like terms, it does not achieve the same result as distributing a binomial across a trinomial.

Option B: 2x(3x^2 + 4x - 7)

Option B multiplies the expression $(3x^2 + 4x - 7)$ by $2x$. While this is a valid application of the distributive property, it doesn't account for the full multiplication by $(x - 2)$. It only considers the 2x part and omits the -2 term. Therefore, option B is also incorrect. To further clarify, this option only distributes 2x across the trinomial, effectively scaling the expression by a quadratic term. It fails to incorporate the subtraction element of the binomial (x - 2), which is crucial for achieving equivalence. The resulting expression would only represent a portion of the expanded form, lacking the terms generated by multiplying with the -2. This makes it an incomplete representation of the original expression's expansion. Thus, while 2x(3x^2 + 4x - 7) is a valid algebraic operation, it doesn't encapsulate the complete multiplication required by (3x^2 + 4x - 7)(x - 2).

Option C: (3x^2 + 4x - 7)(x) + (3x^2 + 4x - 7)(-2)

This option correctly applies the distributive property. It breaks down the multiplication by $(x - 2)$ into two separate multiplications: one by $x$ and one by $-2$. This aligns perfectly with the distributive property, where $(a)(b + c) = ab + ac$. In our case, $a = (3x^2 + 4x - 7)$, $b = x$, and $c = -2$. This option accurately represents the expansion of the original expression and is therefore the correct answer. This option demonstrates a clear understanding of the distributive property by correctly segregating the multiplication across the binomial. Each term of the trinomial is multiplied by both x and -2, ensuring that all necessary combinations are accounted for. This approach mirrors the expansion process that would be performed manually, making it a direct and accurate representation of the original expression's equivalent form. The structure of this option aligns perfectly with the distributive property's application to binomial multiplication with a trinomial.

Option D: x(3x^2 + 4x - 7) - 2

Option D multiplies $(3x^2 + 4x - 7)$ by $x$ but then simply subtracts $2$. This is incorrect because the $-2$ needs to be multiplied by the entire expression $(3x^2 + 4x - 7)$, not just subtracted as a constant. Therefore, option D is not equivalent. Expanding on why this is incorrect, this option only partially distributes the binomial (x - 2). It correctly multiplies x with the trinomial but fails to distribute the -2 across all the terms within the trinomial. Instead, it subtracts 2 as a constant, which is a fundamentally different operation. The distributive property mandates that -2 should multiply each term of (3x^2 + 4x - 7), resulting in a quadratic expression that is then combined with the cubic expression from the x multiplication. The absence of this complete distribution renders the option non-equivalent to the original expression.

Conclusion

The correct expression equivalent to $(3x^2 + 4x - 7)(x - 2)$ is Option C: $(3x^2 + 4x - 7)(x) + (3x^2 + 4x - 7)(-2)$. This option demonstrates a proper understanding and application of the distributive property, ensuring that each term is multiplied correctly to expand the expression. Understanding the distributive property is not just about expanding expressions; it's also about recognizing equivalent forms. By mastering this concept, you can simplify complex algebraic manipulations and solve a wider range of problems with confidence.

To further solidify our understanding, let's manually expand the expression $(3x^2 + 4x - 7)(x - 2)$ using the distributive property. This process will give us a concrete example of how the correct option, Option C, accurately represents the expanded form.

Step 1: Distribute x

First, we multiply each term in $(3x^2 + 4x - 7)$ by $x$:

• $x * 3x^2 = 3x^3$
• $x * 4x = 4x^2$
• $x * -7 = -7x$

This gives us the expression $3x^3 + 4x^2 - 7x$.

Step 2: Distribute -2

Next, we multiply each term in $(3x^2 + 4x - 7)$ by $-2$:

• $-2 * 3x^2 = -6x^2$
• $-2 * 4x = -8x$
• $-2 * -7 = 14$

This gives us the expression $-6x^2 - 8x + 14$.

Step 3: Combine the Results

Now, we combine the results from Step 1 and Step 2:

$(3x^3 + 4x^2 - 7x) + (-6x^2 - 8x + 14)$

Step 4: Simplify by Combining Like Terms

Finally, we combine like terms:

• $3x^3$ (no other cubic terms)
• $4x^2 - 6x^2 = -2x^2$
• $-7x - 8x = -15x$
• $14$ (constant term)

This results in the simplified expression $3x^3 - 2x^2 - 15x + 14$.

Comparing the Expanded Form to Option C

Option C states: $(3x^2 + 4x - 7)(x) + (3x^2 + 4x - 7)(-2)$

This is precisely the process we followed in Steps 1 and 2. Option C represents the intermediate step where we have distributed both $x$ and $-2$ but haven't yet combined the terms. Our manual expansion confirms that Option C is indeed the equivalent expression before simplification.

Why Other Options Fail

Let's briefly revisit why the other options are incorrect in light of our step-by-step expansion:

• Option A: $(3x^2 + 4x - 7) + 2(3x^2 + 4x - 7)$ is equivalent to $3(3x^2 + 4x - 7)$, which would result in a quadratic expression, not the cubic expression we obtained after expansion.
• Option B: $2x(3x^2 + 4x - 7)$ only accounts for the multiplication by $2x$, missing the multiplication by $-2$. This results in $6x^3 + 8x^2 - 14x$, which is not the complete expanded form.
• Option D: $x(3x^2 + 4x - 7) - 2$ multiplies by $x$ correctly but then only subtracts $2$ as a constant, failing to distribute the $-2$ across the entire expression. This gives us $3x^3 + 4x^2 - 7x - 2$, which is also incorrect.

Real-World Applications of Equivalent Expressions

The concept of equivalent expressions is not just an abstract mathematical idea; it has practical applications in various fields. Understanding how to manipulate and simplify expressions is crucial in fields such as physics, engineering, computer science, and economics.

1. Physics

In physics, many formulas involve complex algebraic expressions. For instance, kinematic equations, which describe the motion of objects, often require simplification to solve for specific variables. Being able to recognize and manipulate equivalent expressions allows physicists to simplify equations and make calculations more manageable. Consider the equation for the distance traveled by an object under constant acceleration:

d = v_0t + rac{1}{2}at^2

If we need to solve for the initial velocity $v_0$, we would manipulate the equation to isolate $v_0$. This involves recognizing equivalent forms of the equation by subtracting rac{1}{2}at^2 from both sides.

2. Engineering

Engineers frequently work with complex systems that are modeled using algebraic equations. Whether it's designing electrical circuits, mechanical systems, or chemical processes, engineers need to simplify and manipulate equations to optimize performance, predict behavior, and troubleshoot issues. For example, in electrical engineering, the total resistance in a parallel circuit can be calculated using:

rac{1}{R_{total}} = rac{1}{R_1} + rac{1}{R_2} + ... + rac{1}{R_n}

Simplifying this equation to solve for $R_{total}$ or one of the individual resistances $R_i$ requires a strong understanding of equivalent expressions and algebraic manipulation.

3. Computer Science

In computer science, particularly in algorithm design and optimization, manipulating algebraic expressions is essential. Algorithms often involve mathematical operations, and simplifying these operations can lead to more efficient code. For instance, in cryptography, complex mathematical expressions are used to encrypt and decrypt data. Understanding the equivalent forms of these expressions can help in designing more secure and efficient cryptographic systems. Additionally, in areas like machine learning, feature scaling and data normalization often involve transforming data using algebraic expressions, and recognizing equivalent forms can help in optimizing these processes.

4. Economics

Economic models often use algebraic equations to represent relationships between different economic variables. For example, supply and demand curves, cost functions, and revenue functions are all expressed algebraically. Economists use these models to analyze market behavior, predict economic trends, and evaluate policy interventions. Simplifying these equations and recognizing equivalent forms allows economists to derive insights and make predictions more effectively. For instance, the profit function $\pi = TR - TC$, where $TR$ is total revenue and $TC$ is total cost, can be further expanded and manipulated depending on the specific forms of the revenue and cost functions.

Tips for Mastering Equivalent Expressions

Mastering the concept of equivalent expressions requires practice and a solid understanding of algebraic principles. Here are some tips to help you improve your skills:

1. Understand the Basic Properties: Make sure you have a firm grasp of the commutative, associative, and distributive properties. These properties are the foundation for manipulating algebraic expressions.
2. Practice Regularly: Like any mathematical skill, practice is key. Work through a variety of problems involving different types of expressions.
3. Use Visual Aids: Visual aids like diagrams or color-coding can help you keep track of terms and operations when expanding or simplifying expressions.
4. Check Your Work: Always double-check your work by substituting numerical values for the variables. If two expressions are equivalent, they should yield the same result for any value of the variables.
5. Break Down Complex Problems: Break down complex problems into smaller, more manageable steps. This makes the process less daunting and reduces the likelihood of errors.
6. Seek Help When Needed: Don't hesitate to ask for help from teachers, tutors, or classmates if you're struggling with a particular concept or problem.

In conclusion, understanding equivalent expressions is a fundamental skill in algebra with wide-ranging applications. By mastering the distributive property and practicing regularly, you can confidently manipulate and simplify algebraic expressions. The correct option for the given question is Option C, which accurately represents the expanded form of $(3x^2 + 4x - 7)(x - 2)$. Remember to apply the step-by-step expansion process and compare the results to the provided options to ensure accuracy. This skill will not only help you succeed in mathematics but also in various fields that rely on algebraic modeling and problem-solving.