Which Statement Is True About The Polynomial $3x^2y^2 - 5xy^2 - 3x^2y^2$?A. It Has 2 Terms And A Degree Of 2.B. It Has 2 Terms And A Degree Of 3.C. It Has 4 Terms And A Degree Of 2.D. It Has 4 Terms And A Degree Of 4.

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Introduction

Polynomials are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the world of polynomials and examine a given statement to determine its validity. The statement in question is about the polynomial 3x2y2−5xy2−3x2y23x^2y^2 - 5xy^2 - 3x^2y^2. We will analyze each term and determine the degree of the polynomial, as well as the number of terms it contains.

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as x, y, or z, and the coefficients are numbers that are multiplied with the variables. Polynomials can be classified based on their degree, which is the highest power of the variable in the polynomial.

Analyzing the Given Polynomial

The given polynomial is 3x2y2−5xy2−3x2y23x^2y^2 - 5xy^2 - 3x^2y^2. To determine the degree of the polynomial, we need to examine each term and identify the highest power of the variable.

Term 1: 3x2y23x^2y^2

The first term is 3x2y23x^2y^2. In this term, the variable x has a power of 2, and the variable y has a power of 2. Therefore, the degree of this term is 4.

Term 2: −5xy2-5xy^2

The second term is −5xy2-5xy^2. In this term, the variable x has a power of 1, and the variable y has a power of 2. Therefore, the degree of this term is 3.

Term 3: −3x2y2-3x^2y^2

The third term is −3x2y2-3x^2y^2. In this term, the variable x has a power of 2, and the variable y has a power of 2. Therefore, the degree of this term is 4.

Determining the Degree of the Polynomial

Now that we have analyzed each term, we can determine the degree of the polynomial. The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power of the variable is 4, which is the degree of the polynomial.

Determining the Number of Terms

We also need to determine the number of terms in the polynomial. A term is a single part of the polynomial, separated by addition or subtraction. In this case, we have three terms: 3x2y23x^2y^2, −5xy2-5xy^2, and −3x2y2-3x^2y^2. Therefore, the polynomial has 3 terms.

Conclusion

In conclusion, the given statement is false. The polynomial 3x2y2−5xy2−3x2y23x^2y^2 - 5xy^2 - 3x^2y^2 has 3 terms, not 2, and a degree of 4, not 2 or 3.

Answer

The correct answer is D. It has 3 terms and a degree of 4.

Final Thoughts

Understanding polynomials is a crucial aspect of mathematics, and being able to analyze and solve polynomial equations is essential for solving various mathematical problems. In this article, we examined a given statement about a polynomial and determined its validity. We hope that this article has provided valuable insights into the world of polynomials and has helped readers to better understand this fundamental concept in mathematics.

Additional Resources

For those who want to learn more about polynomials, here are some additional resources:

  • Khan Academy: Polynomials
  • Mathway: Polynomials
  • Wolfram MathWorld: Polynomials

Conclusion

Introduction

In our previous article, we explored the concept of polynomials and analyzed a given statement about a polynomial. In this article, we will answer some frequently asked questions about polynomials, providing a deeper understanding of this fundamental concept in mathematics.

Q: What is a polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as x, y, or z, and the coefficients are numbers that are multiplied with the variables.

Q: What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial x2+3x+2x^2 + 3x + 2, the degree is 2 because the highest power of the variable x is 2.

Q: How do I determine the degree of a polynomial?

To determine the degree of a polynomial, you need to examine each term and identify the highest power of the variable. You can do this by looking at the exponent of the variable in each term.

Q: What is the difference between a polynomial and a rational expression?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression, on the other hand, is an expression that can be written as the ratio of two polynomials.

Q: Can a polynomial have a negative degree?

No, a polynomial cannot have a negative degree. The degree of a polynomial is always a non-negative integer.

Q: How do I add and subtract polynomials?

To add and subtract polynomials, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, in the polynomial x2+3x+2x^2 + 3x + 2 and x2+2x+1x^2 + 2x + 1, the like terms are x2x^2 and 3x3x.

Q: Can a polynomial have a variable with a fractional exponent?

No, a polynomial cannot have a variable with a fractional exponent. The exponent of a variable in a polynomial must be a non-negative integer.

Q: How do I multiply polynomials?

To multiply polynomials, you need to use the distributive property. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac.

Q: Can a polynomial have a variable with a negative exponent?

No, a polynomial cannot have a variable with a negative exponent. The exponent of a variable in a polynomial must be a non-negative integer.

Q: What is the difference between a polynomial and an algebraic expression?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression, on the other hand, is a general term that refers to any expression that can be written using variables, coefficients, and mathematical operations.

Conclusion

In conclusion, polynomials are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. We hope that this article has provided valuable insights into the world of polynomials and has helped readers to better understand this fundamental concept in mathematics.

Additional Resources

For those who want to learn more about polynomials, here are some additional resources:

  • Khan Academy: Polynomials
  • Mathway: Polynomials
  • Wolfram MathWorld: Polynomials

Final Thoughts

Understanding polynomials is a crucial aspect of mathematics, and being able to analyze and solve polynomial equations is essential for solving various mathematical problems. We hope that this article has provided valuable insights into the world of polynomials and has helped readers to better understand this fundamental concept in mathematics.