Simplifying Polynomials Finding The Simplest Form Of -(2x^3 + X^2) + 3(x^3 - 4x^2)

In the realm of algebra, simplifying polynomial expressions is a fundamental skill. It involves combining like terms and applying the distributive property to reduce an expression to its simplest form. This ability is crucial for solving equations, graphing functions, and tackling more advanced mathematical concepts. In this article, we will delve into the process of simplifying the expression $-(2x^3 + x^2) + 3(x^3 - 4x^2)$, providing a detailed, step-by-step explanation that will empower you to confidently handle similar problems.

Understanding Polynomials

Before we dive into the simplification process, let's briefly revisit what polynomials are. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Each term in a polynomial is a product of a constant (coefficient) and one or more variables raised to non-negative integer powers. For instance, in the expression $-(2x^3 + x^2) + 3(x^3 - 4x^2)$, we have terms like $-2x^3$, $-x^2$, $3x^3$, and $-12x^2$. The coefficients are -2, -1, 3, and -12, respectively, and the variable 'x' is raised to the powers of 3 and 2.

Polynomial expressions are the building blocks of many algebraic concepts, and mastering their simplification is essential for success in mathematics. Understanding the structure and components of polynomials allows us to effectively manipulate and simplify them. When simplifying polynomials, the goal is to combine like terms, which are terms that have the same variable raised to the same power. This process allows us to write the polynomial in a more concise and manageable form. For example, $2x^3$ and $3x^3$ are like terms because they both have the variable 'x' raised to the power of 3. On the other hand, $x^2$ and $x^3$ are not like terms because they have different powers of 'x'.

Step 1: Applying the Distributive Property

The first step in simplifying the expression $-(2x^3 + x^2) + 3(x^3 - 4x^2)$ is to apply the distributive property. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our expression, we have two instances where we need to apply the distributive property:

1. The negative sign in front of the parenthesis $-(2x^3 + x^2)$ can be treated as -1 multiplied by the expression inside the parenthesis. So, we distribute -1 to both terms inside the parenthesis: -1 * (2x^3) = -2x^3 and -1 * (x^2) = -x^2.
2. Similarly, we have 3 multiplied by the expression inside the second parenthesis: 3(x^3 - 4x^2). We distribute 3 to both terms inside the parenthesis: 3 * (x^3) = 3x^3 and 3 * (-4x^2) = -12x^2.

Applying the distributive property, the expression becomes: $-2x^3 - x^2 + 3x^3 - 12x^2$. This step is crucial because it removes the parentheses, allowing us to combine like terms in the next step. The distributive property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations. Mastering this property is essential for success in algebra and beyond. When applying the distributive property, it is important to pay close attention to the signs of the terms. A negative sign in front of a parenthesis changes the sign of each term inside the parenthesis.

Step 2: Identifying and Combining Like Terms

After applying the distributive property, our expression is now: $-2x^3 - x^2 + 3x^3 - 12x^2$. The next step is to identify and combine like terms. Remember, like terms are terms that have the same variable raised to the same power. In our expression, we have two pairs of like terms:

1. $-2x^3$ and $3x^3$ are like terms because they both have the variable 'x' raised to the power of 3.
2. $-x^2$ and $-12x^2$ are like terms because they both have the variable 'x' raised to the power of 2.

To combine like terms, we simply add or subtract their coefficients. For the $x^3$ terms, we have -2 + 3 = 1. So, $-2x^3 + 3x^3 = 1x^3$, which is simply written as $x^3$. For the $x^2$ terms, we have -1 - 12 = -13. So, $-x^2 - 12x^2 = -13x^2$.

Combining like terms is a critical step in simplifying polynomial expressions. It allows us to reduce the expression to its most concise form, making it easier to work with. Carefully identifying and combining like terms ensures that the expression is simplified correctly. When combining like terms, it is helpful to rearrange the expression so that like terms are grouped together. This can make it easier to see which terms can be combined.

Step 3: Writing the Simplified Expression

After combining like terms, we have: $x^3 - 13x^2$. This is the simplest form of the expression $-(2x^3 + x^2) + 3(x^3 - 4x^2)$. We have successfully applied the distributive property and combined like terms to arrive at the simplified expression. This final form is much easier to understand and work with compared to the original expression.

The simplified expression, $x^3 - 13x^2$, is a quadratic polynomial. It represents a curve when graphed and has important properties that can be analyzed. The process of simplifying expressions not only makes them easier to work with but also reveals their underlying structure and properties. In this case, simplifying the expression allowed us to identify it as a quadratic polynomial and understand its behavior.

Therefore, the simplest form of the expression $-(2x^3 + x^2) + 3(x^3 - 4x^2)$ is $x^3 - 13x^2$. This corresponds to option B in the given choices.

Conclusion

Simplifying polynomial expressions is a fundamental skill in algebra. By understanding the distributive property and the concept of like terms, you can confidently simplify a wide range of expressions. In this article, we have demonstrated a step-by-step approach to simplifying the expression $-(2x^3 + x^2) + 3(x^3 - 4x^2)$. By following these steps, you can simplify similar expressions and enhance your algebraic skills.

Mastering the art of simplifying polynomial expressions opens doors to more advanced mathematical concepts. The ability to manipulate and simplify expressions is crucial for solving equations, graphing functions, and understanding calculus. By practicing and applying these techniques, you can build a strong foundation in algebra and excel in your mathematical journey.

In summary, simplifying polynomial expressions involves the following key steps:

1. Apply the distributive property to remove parentheses.
2. Identify like terms (terms with the same variable raised to the same power).
3. Combine like terms by adding or subtracting their coefficients.
4. Write the simplified expression in its most concise form.

By consistently applying these steps, you can simplify any polynomial expression with confidence and accuracy. Remember, practice makes perfect, so keep working on these skills to solidify your understanding and build your mathematical proficiency.