When dealing with polynomials in algebra, one fundamental concept is finding the additive inverse of a given polynomial. The additive inverse is a number that, when added to the original polynomial, yields a sum of zero. In this article, we will focus on determining the additive inverse of the polynomial –9xy² + 6x²y – 5x³. We will provide detailed explanations and guide you through the process, ensuring that you grasp this concept fully.

## Finding the Additive Inverse of the Polynomial –9xy² + 6x²y – 5x³

To find the additive inverse of the polynomial –9xy² + 6x²y – 5x³, we will follow a systematic approach that involves reversing the signs of each term in the polynomial.

### Understanding the Additive Inverse

Before diving into the specific polynomial, let’s briefly revisit the concept of the additive inverse. The additive inverse of a number ‘a’ is denoted as ‘-a,’ and it satisfies the property: a + (-a) = 0. This principle extends to polynomials as well, where we find the additive inverse by reversing the sign of each term.

### Identifying the Terms in the Polynomial

To proceed, let’s break down the given polynomial into its individual terms:

Term 1: -9xy²

Term 2: 6x²y

Term 3: -5x³

### Finding the Additive Inverse of Each Term

Now that we have identified the terms, let’s find their respective additive inverses:

Additive Inverse of Term 1: -(-9xy²) = 9xy²

Additive Inverse of Term 2: -(6x²y) = -6x²y

Additive Inverse of Term 3: -(-5x³) = 5x³

### Combining the Additive Inverses

After determining the additive inverse of each term, we combine them to form the final additive inverse of the entire polynomial:

Additive Inverse of –9xy² + 6x²y – 5x³ = 9xy² – 6x²y + 5x³

The additive inverse of the polynomial –9xy² + 6x²y – 5x³ is 9xy² – 6x²y + 5x³.

## Solving Equations Using the Additive Inverse

The concept of the additive inverse plays a crucial role in solving equations involving polynomials. Let’s explore some examples to illustrate its application.

### Example 1: Solving for x in 4x² – 7xy + 2y² = 0

To solve this quadratic equation, we first need to isolate the terms involving ‘x.’ Using the additive inverse, we can rewrite the equation as follows:

4x² – 7xy + 2y² + (-4x²) = 0 + (-4x²)

Simplifying the equation, we get:

-3x² – 7xy + 2y² = -4x²

Now, let’s isolate ‘x’ further:

-3x² + 4x² = 7xy – 2y²

x² = 7xy – 2y²

Finally, we find ‘x’ by taking the square root of both sides:

x = ±√(7xy – 2y²)

### Example 2: Solving for y in 3x³ – 2xy + 5y = 7

To solve this cubic equation for ‘y,’ we follow a similar approach:

3x³ – 2xy + 5y – 7 + (-5y) = 7 + (-5y)

Simplifying the equation, we get:

3x³ – 2xy = 12 – 5y

Now, let’s isolate ‘y’:

3x³ – 2xy + 5y = 12

5y = 12 – 3x³ + 2xy

y = (12 – 3x³ + 2xy) / 5

## Common FAQs about the Additive Inverse of Polynomials

### What is the definition of the additive inverse of a polynomial?

The additive inverse of a polynomial is a polynomial with reversed signs for each term of the original polynomial. When added together, the original polynomial and its additive inverse yield a sum of zero.

### How do you find the additive inverse of a polynomial?

To find the additive inverse of a polynomial, reverse the sign of each term within the polynomial. Combine the terms with their reversed signs to obtain the final additive inverse.

### What is the additive inverse of the polynomial –3x + 5x² – 2x³?

To find the additive inverse of the given polynomial –3x + 5x² – 2x³, reverse the sign of each term:

Additive Inverse of –3x + 5x² – 2x³ = 3x – 5x² + 2x³

### Can a polynomial and its additive inverse be equal?

Yes, a polynomial and its additive inverse can be equal when both have all terms equal to zero. In such cases, adding the polynomial and its additive inverse will yield a sum of zero.

### How does the additive inverse help in solving equations?

The additive inverse helps in solving equations by simplifying them and isolating variables. By adding the additive inverse to both sides of an equation, we can eliminate certain terms and make solving for variables easier.

### Does the additive inverse work for polynomials of any degree?

Yes, the concept of the additive inverse applies to polynomials of any degree. Regardless of the degree of the polynomial, reversing the sign of each term results in its additive inverse.

## Conclusion

Understanding the additive inverse of polynomials is a fundamental concept in algebra. In this article, we have explored how to find the additive inverse of the polynomial –9xy² + 6x²y – 5x³ through step-by-step explanations and examples. Additionally, we demonstrated how the additive inverse can be applied to solve equations involving polynomials. By mastering this concept, you gain a valuable tool to simplify complex polynomial expressions and equations.