What Is The Additive Inverse Of The Polynomial?A. \[$-9xy^2 - 6x^2y + 5x^3\$\]B. \[$-9xy^2 - 6x^2y - 5x^3\$\]C. \[$9xy^2 + 6x^2y + 5x^3\$\]D. \[$8xy^2 - 6x^2y + 5x^3\$\]

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Introduction to Polynomials and Additive Inverse

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a zero polynomial. In other words, it is the opposite of the polynomial, which when added together, cancels out the terms.

What is the Additive Inverse of a Polynomial?

The additive inverse of a polynomial is found by changing the sign of each term in the polynomial. This means that if we have a polynomial with a positive coefficient, we change it to a negative coefficient, and vice versa. For example, if we have the polynomial 3x2+2x−43x^2 + 2x - 4, the additive inverse would be −3x2−2x+4-3x^2 - 2x + 4.

Finding the Additive Inverse of a Polynomial

To find the additive inverse of a polynomial, we need to follow these steps:

  1. Identify the terms: Break down the polynomial into its individual terms.
  2. Change the sign: Change the sign of each term in the polynomial.
  3. Combine like terms: Combine any like terms that may have resulted from changing the sign.

Example 1: Finding the Additive Inverse of a Polynomial

Let's find the additive inverse of the polynomial −9xy2−6x2y+5x3-9xy^2 - 6x^2y + 5x^3.

  1. Identify the terms: The terms in the polynomial are −9xy2-9xy^2, −6x2y-6x^2y, and 5x35x^3.
  2. Change the sign: Change the sign of each term in the polynomial. This gives us 9xy29xy^2, 6x2y6x^2y, and −5x3-5x^3.
  3. Combine like terms: There are no like terms in this polynomial, so the additive inverse is simply 9xy2+6x2y−5x39xy^2 + 6x^2y - 5x^3.

Example 2: Finding the Additive Inverse of a Polynomial

Let's find the additive inverse of the polynomial −9xy2−6x2y−5x3-9xy^2 - 6x^2y - 5x^3.

  1. Identify the terms: The terms in the polynomial are −9xy2-9xy^2, −6x2y-6x^2y, and −5x3-5x^3.
  2. Change the sign: Change the sign of each term in the polynomial. This gives us 9xy29xy^2, 6x2y6x^2y, and 5x35x^3.
  3. Combine like terms: There are no like terms in this polynomial, so the additive inverse is simply 9xy2+6x2y+5x39xy^2 + 6x^2y + 5x^3.

Conclusion

In conclusion, the additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a zero polynomial. To find the additive inverse of a polynomial, we need to change the sign of each term in the polynomial and combine like terms. By following these steps, we can find the additive inverse of any polynomial.

Answer

The correct answer is C. 9xy2+6x2y+5x39xy^2 + 6x^2y + 5x^3.

Key Takeaways

  • The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a zero polynomial.
  • To find the additive inverse of a polynomial, we need to change the sign of each term in the polynomial and combine like terms.
  • The additive inverse of a polynomial is found by changing the sign of each term in the polynomial.

Frequently Asked Questions

  • What is the additive inverse of a polynomial? The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a zero polynomial.
  • How do I find the additive inverse of a polynomial? To find the additive inverse of a polynomial, we need to change the sign of each term in the polynomial and combine like terms.
  • What is the difference between the additive inverse and the negative of a polynomial? The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a zero polynomial. The negative of a polynomial is a polynomial with the opposite sign of each term.

Q1: What is the additive inverse of a polynomial?

A1: The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a zero polynomial. In other words, it is the opposite of the polynomial, which when added together, cancels out the terms.

Q2: How do I find the additive inverse of a polynomial?

A2: To find the additive inverse of a polynomial, you need to follow these steps:

  1. Identify the terms: Break down the polynomial into its individual terms.
  2. Change the sign: Change the sign of each term in the polynomial.
  3. Combine like terms: Combine any like terms that may have resulted from changing the sign.

Q3: What is the difference between the additive inverse and the negative of a polynomial?

A3: The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, results in a zero polynomial. The negative of a polynomial is a polynomial with the opposite sign of each term. For example, the additive inverse of the polynomial 3x2+2x−43x^2 + 2x - 4 is −3x2−2x+4-3x^2 - 2x + 4, while the negative of the polynomial is −3x2−2x+4-3x^2 - 2x + 4.

Q4: Can I find the additive inverse of a polynomial with variables?

A4: Yes, you can find the additive inverse of a polynomial with variables. The process is the same as finding the additive inverse of a polynomial with constants. For example, the additive inverse of the polynomial 2xy2+3x2y−4x32xy^2 + 3x^2y - 4x^3 is −2xy2−3x2y+4x3-2xy^2 - 3x^2y + 4x^3.

Q5: Can I find the additive inverse of a polynomial with fractions?

A5: Yes, you can find the additive inverse of a polynomial with fractions. The process is the same as finding the additive inverse of a polynomial with constants. For example, the additive inverse of the polynomial 12x2+34x−13\frac{1}{2}x^2 + \frac{3}{4}x - \frac{1}{3} is −12x2−34x+13-\frac{1}{2}x^2 - \frac{3}{4}x + \frac{1}{3}.

Q6: Can I find the additive inverse of a polynomial with exponents?

A6: Yes, you can find the additive inverse of a polynomial with exponents. The process is the same as finding the additive inverse of a polynomial with constants. For example, the additive inverse of the polynomial 2x3+3x2−4x2x^3 + 3x^2 - 4x is −2x3−3x2+4x-2x^3 - 3x^2 + 4x.

Q7: Can I find the additive inverse of a polynomial with radicals?

A7: Yes, you can find the additive inverse of a polynomial with radicals. The process is the same as finding the additive inverse of a polynomial with constants. For example, the additive inverse of the polynomial 2x+3x2−4x32\sqrt{x} + 3\sqrt{x^2} - 4\sqrt{x^3} is −2x−3x2+4x3-2\sqrt{x} - 3\sqrt{x^2} + 4\sqrt{x^3}.

Q8: Can I find the additive inverse of a polynomial with complex numbers?

A8: Yes, you can find the additive inverse of a polynomial with complex numbers. The process is the same as finding the additive inverse of a polynomial with constants. For example, the additive inverse of the polynomial 2i+3i2−4i32i + 3i^2 - 4i^3 is −2i−3i2+4i3-2i - 3i^2 + 4i^3.

Q9: Can I find the additive inverse of a polynomial with multiple variables?

A9: Yes, you can find the additive inverse of a polynomial with multiple variables. The process is the same as finding the additive inverse of a polynomial with one variable. For example, the additive inverse of the polynomial 2xy2+3x2y−4x32xy^2 + 3x^2y - 4x^3 is −2xy2−3x2y+4x3-2xy^2 - 3x^2y + 4x^3.

Q10: Can I find the additive inverse of a polynomial with a constant term?

A10: Yes, you can find the additive inverse of a polynomial with a constant term. The process is the same as finding the additive inverse of a polynomial without a constant term. For example, the additive inverse of the polynomial 2x2+3x−42x^2 + 3x - 4 is −2x2−3x+4-2x^2 - 3x + 4.

Conclusion

In conclusion, finding the additive inverse of a polynomial is a straightforward process that involves changing the sign of each term in the polynomial and combining like terms. By following these steps, you can find the additive inverse of any polynomial, regardless of the complexity of the polynomial.