Question

which terms could be used as the first term of the given expression to create a polynomial written in standard form? choose five correct answers. ___ + 8r^{2}s^{4} - 3r^{3}s^{3} the options are: -6r^{5}s^{5}, 3r^{4}s^{5}, \frac{5s^{7}}{6}, -r^{4}s^{5}, s^{5}, \frac{4r}{s^{6}}

Explanation:

Step1: Define standard polynomial form

A polynomial in standard form is ordered by descending degree of each term, where the degree of a term with variables $r$ and $s$ is the sum of their exponents: $\text{degree} = \text{exponent of } r + \text{exponent of } s$.

Step2: Calculate degrees of given terms

• For $8r^2s^4$: $2+4=6$
• For $-3r^3s^3$: $3+3=6$

Step3: Calculate degrees of candidate terms

1. $-6rs^5$: $1+5=6$
2. $-r^4s^5$: $4+5=9$
3. $3r^4s^5$: $4+5=9$
4. $s^5$: $0+5=5$
5. $\frac{5s^7}{6}$: $0+7=7$
6. $\frac{4r}{s^5}=4rs^{-5}$: This is not a polynomial term (negative exponent on $s$).

Step4: Identify valid leading terms

A leading term must:

1. Be a polynomial term (no negative exponents).
2. Have a degree greater than or equal to 6 (the highest degree of existing terms) to be first in standard form.

Answer:

$-r^4s^5$, $3r^4s^5$, $\frac{5s^7}{6}$, $-6rs^5$, $8r^2s^4$
(Note: $-6rs^5$ and $8r^2s^4$ have the same degree as the existing highest-degree terms, so they can also serve as leading terms in standard form alongside the higher-degree terms $-r^4s^5$, $3r^4s^5$, and $\frac{5s^7}{6}$)