Additive Inverse Of Polynomial $-9xy^2 + 6x^2y - 5x^3$

The additive inverse of a polynomial is a fundamental concept in algebra. In this comprehensive guide, we will delve into the process of finding the additive inverse of the polynomial $-9xy^2 + 6x^2y - 5x^3$. We will begin by defining the additive inverse, then explore the steps involved in determining it for a given polynomial, and finally, apply these steps to our specific example. Understanding the concept of additive inverses is crucial for simplifying expressions, solving equations, and performing various algebraic manipulations. Let's embark on this journey to master the additive inverse of polynomials.

Understanding Additive Inverses

To effectively determine the additive inverse of a polynomial, it's essential to first grasp the core concept of additive inverses in mathematics. The additive inverse of any number is the value that, when added to the original number, results in zero. This principle extends to polynomials as well. In simpler terms, the additive inverse of a polynomial is the polynomial that, when added to the original polynomial, equals zero. This concept is rooted in the properties of real numbers and their opposites. For example, the additive inverse of 5 is -5, because 5 + (-5) = 0. Similarly, the additive inverse of -3 is 3, because -3 + 3 = 0. This idea of 'canceling out' to reach zero is the crux of the additive inverse concept.

Now, when we transition to polynomials, which are algebraic expressions consisting of variables and coefficients, the same principle applies. The additive inverse of a polynomial is obtained by changing the sign of each term in the polynomial. This means that if a term is positive, its additive inverse will be negative, and vice versa. The goal remains the same: when the original polynomial and its additive inverse are combined through addition, all terms should cancel each other out, resulting in a zero polynomial. This understanding forms the basis for the practical steps we'll take to find the additive inverse of $-9xy^2 + 6x^2y - 5x^3$.

Steps to Find the Additive Inverse of a Polynomial

Finding the additive inverse of a polynomial is a straightforward process that involves a simple sign change operation. By following these steps, you can easily determine the additive inverse of any given polynomial.

1. Identify Each Term: The first step is to carefully identify each term within the polynomial. A term is a single algebraic expression that is separated from other terms by addition or subtraction signs. For example, in the polynomial $-9xy^2 + 6x^2y - 5x^3$, the terms are $-9xy^2$, $6x^2y$, and $-5x^3$. Understanding this separation is crucial for applying the next step correctly.

2. Change the Sign of Each Term: This is the core of the process. For each term you've identified, change its sign. If a term is positive, make it negative, and if it's negative, make it positive. For instance, if we have the term $-9xy^2$, its sign changes from negative to positive, resulting in $9xy^2$. Similarly, $6x^2y$ becomes $-6x^2y$, and $-5x^3$ becomes $5x^3$. This sign change is based on the additive inverse principle, where a number plus its inverse equals zero.

3. Combine the Terms: Once you have changed the sign of each term, combine them to form the new polynomial. This new polynomial is the additive inverse of the original polynomial. Using our previous examples, the terms $9xy^2$, $-6x^2y$, and $5x^3$ are combined to form the additive inverse polynomial $9xy^2 - 6x^2y + 5x^3$. This resulting polynomial, when added to the original, should yield zero, confirming that it is indeed the additive inverse.

These steps provide a systematic way to find the additive inverse, ensuring accuracy and clarity in your algebraic manipulations. Let's now apply these steps to the polynomial $-9xy^2 + 6x^2y - 5x^3$ to find its additive inverse.

Finding the Additive Inverse of $-9xy^2 + 6x^2y - 5x^3$

Now, let's apply the steps we've outlined to find the additive inverse of the polynomial $-9xy^2 + 6x^2y - 5x^3$. This process will demonstrate how to systematically change the signs of each term and combine them to form the additive inverse.

1. Identify Each Term: The polynomial $-9xy^2 + 6x^2y - 5x^3$ consists of three terms: $-9xy^2$, $6x^2y$, and $-5x^3$. These terms are separated by the addition and subtraction signs within the expression. Recognizing these individual components is the foundation for accurately applying the additive inverse transformation.

2. Change the Sign of Each Term: Next, we change the sign of each term. The term $-9xy^2$ has a negative sign, so we change it to positive, resulting in $9xy^2$. The term $6x^2y$ is positive, so we change it to negative, giving us $-6x^2y$. Lastly, the term $-5x^3$ is negative, and we change it to positive, resulting in $5x^3$. This sign-changing operation is the crux of finding the additive inverse, effectively creating the 'opposite' of each term.

3. Combine the Terms: Finally, we combine the sign-changed terms to form the additive inverse polynomial. Combining $9xy^2$, $-6x^2y$, and $5x^3$ gives us the polynomial $9xy^2 - 6x^2y + 5x^3$. This polynomial is the additive inverse of the original polynomial $-9xy^2 + 6x^2y - 5x^3$. To verify, we can add the original polynomial and its additive inverse together, and the result should be zero.

$(-9xy^2 + 6x^2y - 5x^3) + (9xy^2 - 6x^2y + 5x^3) = 0$

This process confirms that $9xy^2 - 6x^2y + 5x^3$ is indeed the additive inverse of $-9xy^2 + 6x^2y - 5x^3$. By meticulously following these steps, we've successfully found the additive inverse, demonstrating the practical application of this algebraic concept.

Verifying the Additive Inverse

To ensure that the polynomial we found, $9xy^2 - 6x^2y + 5x^3$, is indeed the additive inverse of the original polynomial, $-9xy^2 + 6x^2y - 5x^3$, we must perform a verification step. This step involves adding the two polynomials together and confirming that their sum equals zero. This process is a crucial check in algebra, guaranteeing the accuracy of our result.

When we add the two polynomials, we align like terms and combine their coefficients. Like terms are terms that have the same variables raised to the same powers. In this case, $-9xy^2$ and $9xy^2$ are like terms, $6x^2y$ and $-6x^2y$ are like terms, and $-5x^3$ and $5x^3$ are like terms. Adding these pairs of terms involves adding their coefficients while keeping the variable part the same.

So, the addition is performed as follows:

(-9xy^2 + 6x^2y - 5x^3) + (9xy^2 - 6x^2y + 5x^3)

We group the like terms together:

(-9xy^2 + 9xy^2) + (6x^2y - 6x^2y) + (-5x^3 + 5x^3)

Now, we add the coefficients of each group:

0xy^2 + 0x^2y + 0x^3

This simplifies to:

0

The result of the addition is zero, which confirms that $9xy^2 - 6x^2y + 5x^3$ is indeed the additive inverse of $-9xy^2 + 6x^2y - 5x^3$. This verification step is essential in algebra to ensure the correctness of mathematical operations and solutions. It provides confidence in the result and reinforces the understanding of additive inverses.

Conclusion

In summary, we've explored the concept of additive inverses in the context of polynomials and successfully determined the additive inverse of the polynomial $-9xy^2 + 6x^2y - 5x^3$. We began by defining the additive inverse as the polynomial that, when added to the original polynomial, results in zero. Understanding this fundamental principle is key to performing various algebraic operations accurately.

We then outlined a step-by-step process for finding the additive inverse: identifying each term, changing the sign of each term, and combining the terms to form the new polynomial. Applying these steps to our specific example, we found that the additive inverse of $-9xy^2 + 6x^2y - 5x^3$ is $9xy^2 - 6x^2y + 5x^3$. This process showcases how a systematic approach can simplify algebraic manipulations.

To ensure the correctness of our result, we performed a verification step by adding the original polynomial and its additive inverse. The sum was indeed zero, confirming that $9xy^2 - 6x^2y + 5x^3$ is the additive inverse. This verification underscores the importance of checking mathematical solutions to ensure accuracy.

Understanding and applying the concept of additive inverses is crucial for various algebraic tasks, including simplifying expressions, solving equations, and manipulating polynomials. This exploration provides a solid foundation for further studies in algebra and mathematics, empowering you to tackle more complex problems with confidence. The ability to find additive inverses is not just a mathematical skill but also a tool for logical thinking and problem-solving in a broader context.

Therefore, the correct answer is:

C. $9 x y^2-6 x^2 y+5 x^3$