Simplify -4x^2(3x-7) A Step-by-Step Guide

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It's like learning the basic grammar of a language; once you master it, you can communicate complex ideas more effectively. In this comprehensive guide, we will delve into the process of simplifying the expression $-4x^2(3x-7)$. This seemingly simple expression can be a stepping stone to understanding more complex algebraic manipulations. We'll break down the process step by step, ensuring that you not only arrive at the correct answer but also grasp the underlying principles. Understanding how to simplify expressions like these is crucial for various mathematical applications, from solving equations to tackling calculus problems. This guide will not just provide the solution but also offer insights into the 'why' behind each step. We'll explore the distributive property, the rules of exponents, and how to combine like terms. By the end of this guide, you'll be equipped with the knowledge and skills to confidently simplify similar expressions.

The ability to manipulate algebraic expressions is not just about getting the right answer; it's about developing a deeper understanding of mathematical structures. This understanding is crucial for success in higher-level mathematics and related fields. The principles we'll cover here, such as the distributive property and the rules of exponents, are fundamental building blocks for more advanced concepts. Therefore, mastering these basics is an investment in your mathematical future. Whether you're a student preparing for an exam, a professional looking to brush up on your skills, or simply someone with an interest in mathematics, this guide will provide you with a clear and concise explanation of how to simplify algebraic expressions. Let's embark on this journey of mathematical exploration and unlock the power of simplification.

Understanding the Expression

The expression we aim to simplify is $-4x^2(3x-7)$. Before diving into the simplification process, let's dissect the expression. We have a product of two terms: $-4x^2$ and $(3x-7)$. The term $-4x^2$ is a monomial, consisting of a coefficient (-4) and a variable ($x$) raised to a power (2). The term $(3x-7)$ is a binomial, which is an algebraic expression consisting of two terms: $3x$ and -7. The key to simplifying this expression lies in the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In simpler terms, it means we need to multiply the term outside the parentheses ($-4x^2$) by each term inside the parentheses ($3x$ and -7). This is the fundamental principle that will guide our simplification process. Understanding the structure of the expression and the properties that govern it is crucial for successful simplification. Without this understanding, we might make errors or miss opportunities to simplify. Therefore, let's take a moment to appreciate the elegance of the distributive property and its power in simplifying algebraic expressions.

Furthermore, recognizing the components of the expression—the monomials, binomials, coefficients, and variables—is essential for applying the distributive property correctly. Each part plays a role, and understanding their relationships is key to success. For instance, the coefficient -4 in $-4x^2$ needs to be carefully multiplied with the coefficients of the terms inside the parentheses. Similarly, the variable $x^2$ needs to be handled according to the rules of exponents. Therefore, a thorough understanding of the expression's structure is not just a preliminary step; it's an integral part of the simplification process. This initial analysis sets the stage for a smooth and accurate simplification. So, let's proceed with confidence, knowing that we have a solid grasp of the expression we're about to tackle.

Applying the Distributive Property

Now, let's put the distributive property into action. We need to multiply $-4x^2$ by each term inside the parentheses $(3x-7)$. This gives us: $-4x^2 * (3x) + (-4x^2) * (-7)$. Notice how we've carefully distributed the term outside the parentheses to both terms inside. This is a crucial step, as any error here will propagate through the rest of the simplification. Next, we perform the multiplications. Remember that when multiplying terms with the same base (in this case, x), we add their exponents. So, $-4x^2 * (3x)$ becomes $-12x^(2+1)$ which simplifies to $-12x^3$. For the second term, $(-4x^2) * (-7)$, we multiply the coefficients (-4 and -7) to get 28, and the $x^2$ remains unchanged. This gives us $28x^2$. Therefore, after applying the distributive property and performing the multiplications, our expression becomes $-12x^3 + 28x^2$. This step-by-step application of the distributive property is the heart of simplifying this expression. It demonstrates how a single mathematical principle can transform a complex expression into a more manageable form.

Moreover, paying close attention to the signs is paramount. The negative sign in front of $-4x^2$ and the negative sign in -7 both play critical roles in determining the final signs of the terms in the simplified expression. A common mistake is to overlook these signs, leading to an incorrect result. Therefore, double-checking the signs at each step is a good practice. Furthermore, the multiplication of variables with exponents requires careful application of the rules of exponents. Adding the exponents when multiplying terms with the same base is a fundamental rule that must be followed precisely. With these considerations in mind, we can confidently apply the distributive property and move closer to the final simplified expression.

Simplifying the Terms

After applying the distributive property, our expression is now $-12x^3 + 28x^2$. The next step is to examine the terms and see if any further simplification is possible. In this case, we have two terms: $-12x^3$ and $28x^2$. These terms are not