What is true about the completely simplified sum of the polynomials $3 x^2 y^2-2 x y^5$ and $-3 x^2 y^2+3 x^4 y$?
The sum is a trinomial with a degree of 5.
The sum is a trinomial with a degree of 6.

Add the polynomials: ( 3 x 2 y 2 − 2 x y 5 ) + ( − 3 x 2 y 2 + 3 x 4 y ) .
Simplify the sum: − 2 x y 5 + 3 x 4 y .
Determine the degree of each term: The degree of − 2 x y 5 is 6, and the degree of 3 x 4 y is 5.
The degree of the polynomial is 6. The sum is a binomial, not a trinomial. Therefore, the sum has a degree of 6.

Explanation

Understanding the Problem We are given two polynomials, 3 x 2 y 2 − 2 x y 5 and − 3 x 2 y 2 + 3 x 4 y , and we want to find their sum, simplify it, and then determine its properties (number of terms and degree).

Adding and Simplifying the Polynomials First, we add the two polynomials: ( 3 x 2 y 2 − 2 x y 5 ) + ( − 3 x 2 y 2 + 3 x 4 y ) Next, we simplify the sum by combining like terms. Notice that 3 x 2 y 2 and − 3 x 2 y 2 are like terms, and their sum is 0. So we have: 3 x 2 y 2 − 2 x y 5 − 3 x 2 y 2 + 3 x 4 y = ( 3 x 2 y 2 − 3 x 2 y 2 ) − 2 x y 5 + 3 x 4 y = 0 − 2 x y 5 + 3 x 4 y = − 2 x y 5 + 3 x 4 y So the simplified sum is − 2 x y 5 + 3 x 4 y .

Determining the Number of Terms Now, we determine the number of terms in the simplified polynomial. The simplified polynomial is − 2 x y 5 + 3 x 4 y , which has two terms: − 2 x y 5 and 3 x 4 y . Therefore, the simplified sum is a binomial (two terms), not a trinomial (three terms).

Determining the Degree of Each Term Next, we determine the degree of each term in the simplified polynomial. The degree of the term − 2 x y 5 is the sum of the exponents of x and y , which is 1 + 5 = 6 . The degree of the term 3 x 4 y is the sum of the exponents of x and y , which is 4 + 1 = 5 .

Finding the Degree of the Polynomial The degree of the polynomial is the highest degree among all terms. In this case, the degrees of the terms are 6 and 5, so the degree of the polynomial is 6.

Conclusion Since the simplified sum is a binomial with a degree of 6, the statement "The sum is a trinomial with a degree of 5" is false, and the statement "The sum is a trinomial with a degree of 6" is also false. However, if we consider the options given, we need to choose the one that is true. Since the sum is a binomial with degree 6, neither of the options is true. However, the question asks "What is true about the completely simplified sum...", so we need to pick the option that is closest to the truth. The sum has degree 6, so the second option is closer to the truth than the first one.


Examples
Polynomials are used to model various real-world phenomena. For instance, the trajectory of a projectile can be modeled using a quadratic polynomial. Similarly, engineers use polynomials to approximate curves and surfaces in computer-aided design (CAD). Understanding how to add and simplify polynomials is fundamental in these applications, allowing for efficient calculations and accurate predictions.

Answered by GinnyAnswer | 2025-07-07