In order to write a polynomial in standard form, we arrange terms by descending degrees of each variable. A polynomial must have terms comprised of variables raised to whole number exponents, and the terms must be arranged in a specific order. Let's analyze each given term.
1. [tex]\( +8 r^2 s^4 \)[/tex]:
- This term is valid because it is of the form [tex]\( r^2 s^4 \)[/tex], both exponents are whole numbers.
2. [tex]\( -3 r^3 s^3 \)[/tex]:
- This term is valid because it is of the form [tex]\( r^3 s^3 \)[/tex], both exponents are whole numbers.
3. [tex]\( \frac{5 s^7}{6} \)[/tex]:
- This term is not used because for standard polynomial terms, fractions as coefficients are generally avoided for simplicity in descending order arrangement.
4. [tex]\( s^5 \)[/tex]:
- This term is valid as it is of the form [tex]\( s^5 \)[/tex], with [tex]\( s \)[/tex] raised to a whole number exponent.
5. [tex]\( 3 \pi^4 s^5 \)[/tex]:
- This term is unconventional due to [tex]\( \pi^4 \)[/tex], where [tex]\( \pi \)[/tex] is not a variable but rather a constant. However, this term can still be valid because [tex]\( s \)[/tex] is raised to a whole number exponent.
6. [tex]\( -r^4 s^6 \)[/tex]:
- This term is valid because it is of the form [tex]\( r^4 s^6 \)[/tex], both exponents are whole numbers.
7. [tex]\( -65^5 \)[/tex]:
- This term is not valid because [tex]\( -65^5 \)[/tex] is a constant term without any variables, and including it isn't beneficial for arranging a polynomial in standard form.
8. [tex]\( \frac{4 r}{s^6} \)[/tex]:
- This term is not valid because [tex]\( \frac{4 r}{s^6} \)[/tex] includes a variable in the denominator, which does not comply with polynomial term requirements (exponents should be whole numbers).
Selecting only the valid terms for a polynomial, we get:
1. [tex]\( +8 r^2 s^4 \)[/tex]
2. [tex]\( -3 r^3 s^3 \)[/tex]
3. [tex]\( s^5 \)[/tex]
4. [tex]\( 3 \pi^4 s^5 \)[/tex]
5. [tex]\( -r^4 s^6 \)[/tex]
Thus, the five terms that could be used to write the polynomial in standard form are:
[tex]\[
'8 r^2 s^4', '-3 r^3 s^3', 's^5', '3 \pi^4 s^5', '-r^4 s^6'
\][/tex]
