Which Polynomials Are Listed With Their Correct Additive Inverse? Check All That Apply.1. $x^2 + 3x - 2 \ ; \ -x^2 - 3x + 2$2. $-y^7 - 10 \ ; \ Y^7 + 10$3. $6z^5 + 6z^5 - 6z^4 \ ; \ -6z^5 - 6z^5 + 6z^4$4. $x - 1 \ ; \ -x

In mathematics, polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The additive inverse of a polynomial is another polynomial that, when added to the original polynomial, results in a polynomial with a zero constant term. In this article, we will explore which polynomials are listed with their correct additive inverses.

What are Additive Inverses?

The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a polynomial with a zero constant term. For example, the additive inverse of the polynomial $x^2 + 3x - 2$ is $-x^2 - 3x + 2$, because when we add these two polynomials together, we get:

$$x^2 + 3x - 2 + (-x^2 - 3x + 2) = x^2 - x^2 + 3x - 3x - 2 + 2 = 0$$

This means that the additive inverse of $x^2 + 3x - 2$ is indeed $-x^2 - 3x + 2$.

Evaluating the Options

Now, let's evaluate the options given in the problem:

1. $x^2 + 3x - 2 \ ; \ -x^2 - 3x + 2$

As we saw in the previous section, the additive inverse of $x^2 + 3x - 2$ is indeed $-x^2 - 3x + 2$. Therefore, this option is correct.

2. $-y^7 - 10 \ ; \ y^7 + 10$

To find the additive inverse of $-y^7 - 10$, we need to add a polynomial that will result in a zero constant term. Since the constant term is $-10$, we need to add a polynomial with a constant term of $10$. Therefore, the additive inverse of $-y^7 - 10$ is indeed $y^7 + 10$. This option is also correct.

3. $6z^5 + 6z^5 - 6z^4 \ ; \ -6z^5 - 6z^5 + 6z^4$

To find the additive inverse of $6z^5 + 6z^5 - 6z^4$, we need to add a polynomial that will result in a zero constant term. However, this polynomial has no constant term, so its additive inverse is simply the negation of the entire polynomial. Therefore, the additive inverse of $6z^5 + 6z^5 - 6z^4$ is indeed $-6z^5 - 6z^5 + 6z^4$. This option is also correct.

4. $x - 1 \ ; \ -x$

To find the additive inverse of $x - 1$, we need to add a polynomial that will result in a zero constant term. Since the constant term is $-1$, we need to add a polynomial with a constant term of $1$. However, the polynomial $-x$ has no constant term, so it cannot be the additive inverse of $x - 1$. Therefore, this option is incorrect.

Conclusion

In conclusion, the polynomials listed with their correct additive inverses are:

• $x^2 + 3x - 2 \ ; \ -x^2 - 3x + 2$
• $-y^7 - 10 \ ; \ y^7 + 10$
• $6z^5 + 6z^5 - 6z^4 \ ; \ -6z^5 - 6z^5 + 6z^4$

In our previous article, we explored the concept of polynomial additive inverses and identified the correct pairs of polynomials with their additive inverses. In this article, we will answer some frequently asked questions about polynomial additive inverses to help you better understand this concept.

Q: What is the purpose of finding the additive inverse of a polynomial?

A: The purpose of finding the additive inverse of a polynomial is to determine the polynomial that, when added to the original polynomial, results in a polynomial with a zero constant term. This is useful in various mathematical applications, such as solving equations and inequalities.

Q: How do I find the additive inverse of a polynomial?

A: To find the additive inverse of a polynomial, you need to add a polynomial that will result in a zero constant term. This can be done by adding a polynomial with a constant term that is the negative of the constant term of the original polynomial.

Q: What is the difference between a polynomial and its additive inverse?

A: A polynomial and its additive inverse are two polynomials that, when added together, result in a polynomial with a zero constant term. The additive inverse of a polynomial is not the same as the polynomial itself, but rather a polynomial that is added to the original polynomial to result in a zero constant term.

Q: Can a polynomial have more than one additive inverse?

A: No, a polynomial cannot have more than one additive inverse. The additive inverse of a polynomial is unique and is determined by the constant term of the polynomial.

Q: How do I determine if a polynomial is its own additive inverse?

A: A polynomial is its own additive inverse if it has a zero constant term. In this case, the polynomial is equal to its additive inverse, and adding them together results in a polynomial with a zero constant term.

Q: Can a polynomial have a zero additive inverse?

A: No, a polynomial cannot have a zero additive inverse. The additive inverse of a polynomial is a polynomial that is added to the original polynomial to result in a polynomial with a zero constant term. If the polynomial has a zero constant term, then it is its own additive inverse, but it is not a zero additive inverse.

Q: How do I use polynomial additive inverses in real-world applications?

A: Polynomial additive inverses are used in various real-world applications, such as:

• Solving equations and inequalities
• Finding the roots of a polynomial
• Determining the stability of a system
• Analyzing the behavior of a system

Q: What are some common mistakes to avoid when working with polynomial additive inverses?

A: Some common mistakes to avoid when working with polynomial additive inverses include:

• Confusing the polynomial with its additive inverse
• Assuming that a polynomial has more than one additive inverse
• Failing to check if a polynomial has a zero constant term before determining its additive inverse

Conclusion

In conclusion, polynomial additive inverses are an important concept in mathematics that has various real-world applications. By understanding how to find the additive inverse of a polynomial and how to use it in different applications, you can better solve equations and inequalities, find the roots of a polynomial, and analyze the behavior of a system.

Additional Resources

For more information on polynomial additive inverses, you can consult the following resources:

• Polynomial Additive Inverses
• Polynomial Algebra
• Mathematics

Practice Problems

To practice working with polynomial additive inverses, try the following problems:

1. Find the additive inverse of the polynomial $x^2 + 3x - 2$.
2. Determine if the polynomial $x - 1$ is its own additive inverse.
3. Find the additive inverse of the polynomial $-y^7 - 10$.
4. Determine if the polynomial $6z^5 + 6z^5 - 6z^4$ has a zero additive inverse.

Answer Key

1. The additive inverse of the polynomial $x^2 + 3x - 2$ is $-x^2 - 3x + 2$.
2. Yes, the polynomial $x - 1$ is its own additive inverse.
3. The additive inverse of the polynomial $-y^7 - 10$ is $y^7 + 10$.
4. No, the polynomial $6z^5 + 6z^5 - 6z^4$ does not have a zero additive inverse.