To determine which terms can be used as the first term of the polynomial [tex]\(+8r^2s^4 - 3r^3s^3\)[/tex], and maintain the standard form (where terms are arranged in descending order of their degrees), let's examine each of the given options.
A polynomial in standard form is written with terms organized from the highest degree to the lowest degree. The degree of a term is the sum of the exponents of the variables in that term.
Let's examine each option, compute its degree, and see if it can be the first term before [tex]\(8r^2s^4\)[/tex] which has a degree of 6:
1. [tex]\(\frac{55^7}{6}\)[/tex]: This is a numeric fraction with no variables, so it has a degree of 0. Since 0 is less than 6, this term cannot be the first term.
2. [tex]\(s^5\)[/tex]: This term has only one variable, [tex]\(s\)[/tex], with an exponent of 5. Thus, its degree is 5. Since 5 is less than 6, this term cannot be the first term.
3. [tex]\(3r^4s^5\)[/tex]: This term has two variables, [tex]\(r\)[/tex] and [tex]\(s\)[/tex], with exponents 4 and 5 respectively. The sum of the exponents (degree) is [tex]\(4 + 5 = 9\)[/tex]. Since 9 is greater than 6, this term can be the first term.
4. [tex]\(-1.5\)[/tex]: This is a constant, so its degree is 0. Since 0 is less than 6, this term cannot be the first term.
5. [tex]\(-6rs^5\)[/tex]: This term has two variables, [tex]\(r\)[/tex] and [tex]\(s\)[/tex], with exponents 1 and 5 respectively. The sum of the exponents (degree) is [tex]\(1 + 5 = 6\)[/tex]. Since 6 is equal to 6, this term can be the first term.
6. [tex]\(\frac{4r}{5^6}\)[/tex]: This term is [tex]\(\frac{4r}{15625}\)[/tex], which is equivalent to [tex]\(\frac{4}{15625}r\)[/tex]. The degree is the exponent of [tex]\(r\)[/tex], which is 1. Since 1 is less than 6, this term cannot be the first term.
Based on the degrees of the given options, the suitable terms that can be used as the first term of the polynomial to maintain the standard form are:
[tex]\[
\{3r^4s^5, -6rs^5\}
\][/tex]
So the suitable options for the first term are:
- [tex]\(3r^4s^5\)[/tex]
- [tex]\(-6rs^5\)[/tex]
These terms are appropriate because their degrees are equal to or greater than the degree of the term [tex]\(8r^2s^4\)[/tex]. Other options have degrees that are less than 6, making them unsuitable as the first term in the polynomial written in standard form.
