To determine which terms could be used as the first term of the given expression to create a polynomial written in standard form, we need to identify valid polynomial terms and disregard any terms that are not valid or are not written correctly.
A polynomial is an algebraic expression composed of variables and constants, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents on the variables.
Given terms are:
1. [tex]\( +8 r^2 s^4 - 3 r^3 s^3 \)[/tex]
2. [tex]\( \frac{5 s^7}{6} \)[/tex]
3. [tex]\( s^5 \)[/tex]
4. [tex]\( 3 r^4 s^5 \)[/tex]
5. [tex]\( -I^4{ }^6 \)[/tex]
6. [tex]\( -6 r s^5 \)[/tex]
7. [tex]\( \frac{4 r}{s^6}\ )
Let's analyze each term:
1. \( +8 r^2 s^4 - 3 r^3 s^3 \)[/tex] – This is a sum of two polynomial terms: [tex]\(+8 r^2 s^4\)[/tex] and [tex]\(-3 r^3 s^3\)[/tex]. Both are valid polynomial terms.
2. [tex]\( \frac{5 s^7}{6} \)[/tex] – This is a valid polynomial term since it involves variables and a constant coefficient: [tex]\( \frac{5}{6} s^7 \)[/tex].
3. [tex]\( s^5 \)[/tex] – This is a valid polynomial term as it involves only a variable raised to a non-negative integer power.
4. [tex]\( 3 r^4 s^5 \)[/tex] – This is a valid polynomial term as it involves variables raised to non-negative integer powers.
5. [tex]\( -I^4{ }^6 \)[/tex] – This is not a valid polynomial term. [tex]\( I \)[/tex] usually represents the imaginary unit or complex number [tex]\( \sqrt{-1} \)[/tex], and it’s not compatible with the polynomial definition in the context of real numbers.
6. [tex]\( -6 r s^5 \)[/tex] – This is a valid polynomial term as it involves variables raised to non-negative integer powers.
7. [tex]\( \frac{4 r}{s^6} \)[/tex] – This is not a valid polynomial term because it contains a variable in the denominator, making it a non-integer exponent.
Based on this analysis, the valid polynomial terms that could be used to create a polynomial written in standard form are:
1. [tex]\( +8 r^2 s^4 - 3 r^3 s^3 \)[/tex]
2. [tex]\( \frac{5 s^7}{6} \)[/tex]
3. [tex]\( s^5 \)[/tex]
4. [tex]\( 3 r^4 s^5 \)[/tex]
6. [tex]\( -6 r s^5 \)[/tex]
So the correct five options that can be used to create a polynomial are:
- [tex]\( +8 r^2 s^4 - 3 r^3 s^3 \)[/tex]
- [tex]\( \frac{5 s^7}{6} \)[/tex]
- [tex]\( s^5 \)[/tex]
- [tex]\( 3 r^4 s^5 \)[/tex]
- [tex]\( -6 r s^5 \)[/tex]
Therefore, these terms form the valid polynomial choices.
