Answer :
To determine which terms could be used as the first term of the expression to create a polynomial written in standard form, we need to understand the characteristics of polynomial terms in standard form. Such terms involve variables raised to integer powers and potentially have real coefficients.
Here are the given terms we need to consider:
1. [tex]\( 8 r^2 s^4 \)[/tex]
2. [tex]\( -3 r^3 s^3 \)[/tex]
3. [tex]\( \frac{5 s^7}{6} \)[/tex]
4. [tex]\( 5^5 \)[/tex]
5. [tex]\( 3 r^4 s^5 \)[/tex]
6. [tex]\( A_5^6 \)[/tex]
7. [tex]\( -6 r s^5 \)[/tex]
8. [tex]\( \frac{4 r}{5^6} \)[/tex]
We'll evaluate each term one by one to see if they fit the criteria of polynomial terms in standard form.
1. [tex]\( 8 r^2 s^4 \)[/tex]: This term is in standard polynomial form as it involves variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to positive integer powers (2 and 4 respectively) and has a real coefficient (8).
2. [tex]\( -3 r^3 s^3 \)[/tex]: This is also in standard polynomial form because it involves variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to positive integer powers (3 each) and has a real coefficient (-3).
3. [tex]\( \frac{5 s^7}{6} \)[/tex]: This term is in standard polynomial form as well. The variable [tex]\( s \)[/tex] is raised to an integer power (7), and the term has a real coefficient ([tex]\( \frac{5}{6} \)[/tex]).
4. [tex]\( 5^5 \)[/tex]: This is a constant term (3125), which is considered a polynomial term with degree 0.
5. [tex]\( 3 r^4 s^5 \)[/tex]: This term is correctly in standard polynomial form with variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to integer powers (4 and 5 respectively) and a real coefficient (3).
6. [tex]\( A_5^6 \)[/tex]: The notation [tex]\( A_5^6 \)[/tex] suggests a term that is defined outside typical polynomial constructs (it's likely an indexed or non-integer term); hence, it doesn't fit the standard polynomial form.
7. [tex]\( -6 r s^5 \)[/tex]: This term is in standard polynomial form as it has variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to positive integer powers (1 for [tex]\( r \)[/tex] and 5 for [tex]\( s \)[/tex]), and a real coefficient (-6).
8. [tex]\( \frac{4 r}{5^6} \)[/tex]: This term involves a complex fraction but can be rewritten in a simpler form [tex]\( \frac{4}{15625}\cdot r \)[/tex], which means it has a rational coefficient and an integer power variable. Though it can fit into polynomial structure, simpler explanations exclude terms with denominators.
Given the evaluated terms, to form a polynomial in standard form, we can select:
1. [tex]\( 8 r^2 s^4 \)[/tex]
2. [tex]\( -3 r^3 s^3 \)[/tex]
3. [tex]\( 5^5 \)[/tex]
4. [tex]\( 3 r^4 s^5 \)[/tex]
5. [tex]\( -6 r s^5 \)[/tex]
These five terms all meet the criteria for forming a polynomial in standard form:
- [tex]\( 8 r^2 s^4 \)[/tex]
- [tex]\( -3 r^3 s^3 \)[/tex]
- [tex]\( 5^5 \)[/tex]
- [tex]\( 3 r^4 s^5 \)[/tex]
- [tex]\( -6 r s^5 \)[/tex]
Here are the given terms we need to consider:
1. [tex]\( 8 r^2 s^4 \)[/tex]
2. [tex]\( -3 r^3 s^3 \)[/tex]
3. [tex]\( \frac{5 s^7}{6} \)[/tex]
4. [tex]\( 5^5 \)[/tex]
5. [tex]\( 3 r^4 s^5 \)[/tex]
6. [tex]\( A_5^6 \)[/tex]
7. [tex]\( -6 r s^5 \)[/tex]
8. [tex]\( \frac{4 r}{5^6} \)[/tex]
We'll evaluate each term one by one to see if they fit the criteria of polynomial terms in standard form.
1. [tex]\( 8 r^2 s^4 \)[/tex]: This term is in standard polynomial form as it involves variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to positive integer powers (2 and 4 respectively) and has a real coefficient (8).
2. [tex]\( -3 r^3 s^3 \)[/tex]: This is also in standard polynomial form because it involves variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to positive integer powers (3 each) and has a real coefficient (-3).
3. [tex]\( \frac{5 s^7}{6} \)[/tex]: This term is in standard polynomial form as well. The variable [tex]\( s \)[/tex] is raised to an integer power (7), and the term has a real coefficient ([tex]\( \frac{5}{6} \)[/tex]).
4. [tex]\( 5^5 \)[/tex]: This is a constant term (3125), which is considered a polynomial term with degree 0.
5. [tex]\( 3 r^4 s^5 \)[/tex]: This term is correctly in standard polynomial form with variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to integer powers (4 and 5 respectively) and a real coefficient (3).
6. [tex]\( A_5^6 \)[/tex]: The notation [tex]\( A_5^6 \)[/tex] suggests a term that is defined outside typical polynomial constructs (it's likely an indexed or non-integer term); hence, it doesn't fit the standard polynomial form.
7. [tex]\( -6 r s^5 \)[/tex]: This term is in standard polynomial form as it has variables [tex]\( r \)[/tex] and [tex]\( s \)[/tex] raised to positive integer powers (1 for [tex]\( r \)[/tex] and 5 for [tex]\( s \)[/tex]), and a real coefficient (-6).
8. [tex]\( \frac{4 r}{5^6} \)[/tex]: This term involves a complex fraction but can be rewritten in a simpler form [tex]\( \frac{4}{15625}\cdot r \)[/tex], which means it has a rational coefficient and an integer power variable. Though it can fit into polynomial structure, simpler explanations exclude terms with denominators.
Given the evaluated terms, to form a polynomial in standard form, we can select:
1. [tex]\( 8 r^2 s^4 \)[/tex]
2. [tex]\( -3 r^3 s^3 \)[/tex]
3. [tex]\( 5^5 \)[/tex]
4. [tex]\( 3 r^4 s^5 \)[/tex]
5. [tex]\( -6 r s^5 \)[/tex]
These five terms all meet the criteria for forming a polynomial in standard form:
- [tex]\( 8 r^2 s^4 \)[/tex]
- [tex]\( -3 r^3 s^3 \)[/tex]
- [tex]\( 5^5 \)[/tex]
- [tex]\( 3 r^4 s^5 \)[/tex]
- [tex]\( -6 r s^5 \)[/tex]
