What Is True About The Completely Simplified Sum Of The Polynomials 3 X 2 Y 2 − 2 X Y 5 3x^2 Y^2 - 2xy^5 3 X 2 Y 2 − 2 X Y 5 And − 3 X 2 Y 2 + 3 X 4 Y -3x^2 Y^2 + 3x^4 Y − 3 X 2 Y 2 + 3 X 4 Y ?A. The Sum Is A Trinomial With A Degree Of 5.B. The Sum Is A Trinomial With A Degree Of 6.C. The Sum Is A Binomial With A

by ADMIN 315 views

Understanding the Problem

When dealing with polynomials, we often need to add or subtract them to simplify the expression. In this case, we are given two polynomials: 3x2y22xy53x^2 y^2 - 2xy^5 and 3x2y2+3x4y-3x^2 y^2 + 3x^4 y. Our goal is to find the completely simplified sum of these two polynomials.

Adding the Polynomials

To add the polynomials, we need to combine like terms. Like terms are terms that have the same variable and exponent. In this case, we have the following like terms:

  • 3x2y23x^2 y^2 and 3x2y2-3x^2 y^2
  • 2xy5-2xy^5 and 00 (since there is no xy5xy^5 term in the second polynomial)
  • 3x4y3x^4 y and 00 (since there is no x4yx^4 y term in the first polynomial)

Simplifying the Expression

Now, let's simplify the expression by combining the like terms:

3x2y22xy5+(3x2y2)+3x4y3x^2 y^2 - 2xy^5 + (-3x^2 y^2) + 3x^4 y

Using the distributive property, we can rewrite the expression as:

(3x2y23x2y2)2xy5+3x4y(3x^2 y^2 - 3x^2 y^2) - 2xy^5 + 3x^4 y

Now, we can combine the like terms:

02xy5+3x4y0 - 2xy^5 + 3x^4 y

Simplifying further, we get:

2xy5+3x4y-2xy^5 + 3x^4 y

Analyzing the Result

Now that we have simplified the expression, let's analyze the result. The simplified expression is a binomial with two terms: 2xy5-2xy^5 and 3x4y3x^4 y. The degree of a polynomial is the highest power of the variable in any term. In this case, the highest power of xx is 4, and the highest power of yy is 5. Therefore, the degree of the simplified expression is 5.

Conclusion

Based on our analysis, we can conclude that the completely simplified sum of the polynomials 3x2y22xy53x^2 y^2 - 2xy^5 and 3x2y2+3x4y-3x^2 y^2 + 3x^4 y is a binomial with a degree of 5.

Answer Options

Let's examine the answer options:

A. The sum is a trinomial with a degree of 5. B. The sum is a trinomial with a degree of 6. C. The sum is a binomial with a degree of 5.

Based on our analysis, we can see that option C is the correct answer.

Final Answer

The final answer is C.

Understanding Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. They can be classified based on the degree of the polynomial, which is the highest power of the variable in any term.

Frequently Asked Questions

Q: What is the difference between a polynomial and a binomial?

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A binomial, on the other hand, is a polynomial with exactly two terms.

Q: How do I simplify a polynomial?

A: To simplify a polynomial, you need to combine like terms. Like terms are terms that have the same variable and exponent. You can use the distributive property to rewrite the expression and then combine the like terms.

Q: What is the degree of a polynomial?

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

Q: Can a polynomial have a negative degree?

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

Q: How do I add polynomials?

A: To add polynomials, you need to combine like terms. You can use the distributive property to rewrite the expression and then combine the like terms.

Q: Can I subtract polynomials?

A: Yes, you can subtract polynomials. To subtract polynomials, you need to change the sign of each term in the second polynomial and then add the two polynomials.

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

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression, on the other hand, is an algebraic expression consisting of a fraction of two polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator and then cancel out any common factors.

Q: Can I multiply polynomials?

A: Yes, you can multiply polynomials. To multiply polynomials, you need to use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Q: What is the difference between a polynomial and a function?

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A function, on the other hand, is a relation between a set of inputs and a set of possible outputs.

Q: How do I graph a polynomial?

A: To graph a polynomial, you need to find the x-intercepts and y-intercepts of the polynomial and then use a graphing calculator or software to plot the graph.

Conclusion

In this article, we have discussed the basics of polynomials and how to simplify them. We have also answered some frequently asked questions about polynomials and how to work with them.

Final Tips

  • Always combine like terms when simplifying a polynomial.
  • Use the distributive property to rewrite the expression and then combine the like terms.
  • Factor the numerator and denominator when simplifying a rational expression.
  • Use a graphing calculator or software to plot the graph of a polynomial.

Resources

  • Khan Academy: Polynomials
  • Mathway: Polynomials
  • Wolfram Alpha: Polynomials

Related Articles

  • Simplifying Rational Expressions
  • Graphing Polynomials
  • Factoring Polynomials