To determine which terms could be used as the last term of the polynomial expression [tex]\(-5 x^2 y^4 + 9 x^3 y^3\)[/tex] to maintain a standard form, we need to follow the guidelines for writing polynomials in standard form. In standard form, the polynomial terms should be ordered according to the degree of the terms, and within each term, the powers of the variables should be in descending order.
Here is the detailed step-by-step solution:
1. Identify the degrees of the terms in the given polynomial.
- The degree of [tex]\(-5 x^2 y^4\)[/tex] is [tex]\(2 + 4 = 6\)[/tex].
- The degree of [tex]\(9 x^3 y^3\)[/tex] is [tex]\(3 + 3 = 6\)[/tex].
2. Examine each of the potential terms for the correct sequence and order:
- [tex]\(x^5\)[/tex]: The degree is [tex]\(5\)[/tex].
- Since [tex]\(x^5\)[/tex] is a single variable term, it fits as the last term when added to the polynomial.
- [tex]\(y^5\)[/tex]: The degree is [tex]\(5\)[/tex].
- Similar to [tex]\(x^5\)[/tex], as a single variable term, it fits as the last term.
- [tex]\(-4 x^4 y^5\)[/tex]: The degree is [tex]\(4 + 5 = 9\)[/tex].
- This term has the highest degree among the given options and maintains the correct sequence of powers [tex]\(x^4 y^5\)[/tex], placing it rightly in standard form.
- [tex]\(6 x^4 y\)[/tex]: The degree is [tex]\(4 + 1 = 5\)[/tex].
- This term, while correctly ordered, does not fit the pattern of the highest degree or the proper continuation from the given degrees in the polynomial above.
- [tex]\(-x y^5\)[/tex]: The degree is [tex]\(1 + 5 = 6\)[/tex].
- This term is not suitable because it does not follow the highest degree properly when compared to the given polynomial degrees.
- [tex]\(-\frac{x^4}{9}\)[/tex]: This term has a fractional coefficient and does not follow the typical form of polynomials written without fractional coefficients in such standards.
3. Select the terms that can be appropriately added as the last term to maintain standard form:
- [tex]\(x^5\)[/tex]
- [tex]\(y^5\)[/tex]
- [tex]\(-4 x^4 y^5\)[/tex]
Final Answer: The terms that could be used as the last term of the polynomial expression [tex]\(-5 x^2 y^4 + 9 x^3 y^3\)[/tex] to maintain standard form are:
- [tex]\(x^5\)[/tex]
- [tex]\(y^5\)[/tex]
- [tex]\(-4 x^4 y^5\)[/tex]
