Question

what is true about the completely simplified sum of the polynomials $3x^3y^2 - 2xy^5$ and $-3x^3y^2 + 3x^4y$?

• the sum is a trinomial with a degree of 5.
• the sum is a trinomial with a degree of 6.
• the sum is a binomial with a degree of 5.
• the sum is a binomial with a degree of 6.

Explanation:

Step1: Add the polynomials

To add the polynomials \(3x^{3}y^{2}-2xy^{5}\) and \(-3x^{3}y^{2}+3x^{4}y\), we combine like terms.
The like terms for \(x^{3}y^{2}\) are \(3x^{3}y^{2}\) and \(-3x^{3}y^{2}\). Combining them: \(3x^{3}y^{2}- 3x^{3}y^{2}=0\)
The remaining terms are \(-2xy^{5}\) and \(3x^{4}y\). So the sum is \(-2xy^{5}+3x^{4}y\) (or \(3x^{4}y - 2xy^{5}\))

Step2: Determine the number of terms and the degree

• Number of terms: The simplified sum has two terms (\(-2xy^{5}\) and \(3x^{4}y\)), so it is a binomial.
• Degree of a term: For a term \(ax^{m}y^{n}\), the degree is \(m + n\).
• For the term \(-2xy^{5}\), \(m = 1\) and \(n=5\), so the degree is \(1 + 5=6\)
• For the term \(3x^{4}y\), \(m = 4\) and \(n = 1\), so the degree is \(4+1 = 5\)
• The degree of the polynomial is the highest degree of its terms. So the degree of the sum is 6.

Answer:

The sum is a binomial with a degree of 6.