Answer :
To order the polynomials from greatest to least degree, we first need to determine the degree of each polynomial. The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] with a non-zero coefficient in its expression. Let's examine each polynomial:
1. Polynomial: [tex]\( 4x^2 + 5x^8 - 17x^3 - 5x^7 \)[/tex]
- Highest degree: [tex]\( x^8 \)[/tex]
- Degree: 8
2. Polynomial: [tex]\( 23 + x^9 \)[/tex]
- Highest degree: [tex]\( x^9 \)[/tex]
- Degree: 9
3. Polynomial: [tex]\( 3x^5 - 7x^4 + 18x^3 + 20 \)[/tex]
- Highest degree: [tex]\( x^5 \)[/tex]
- Degree: 5
4. Polynomial: [tex]\( 16x^5 - 4x^4 + 23x^3 - 21 + x^6 \)[/tex]
- Highest degree: [tex]\( x^6 \)[/tex]
- Degree: 6
Having found the degrees, we can now list the polynomials from the greatest degree to the least:
1. [tex]\( 23 + x^9 \)[/tex] (Degree 9)
2. [tex]\( 4x^2 + 5x^8 - 17x^3 - 5x^7 \)[/tex] (Degree 8)
3. [tex]\( 16x^5 - 4x^4 + 23x^3 - 21 + x^6 \)[/tex] (Degree 6)
4. [tex]\( 3x^5 - 7x^4 + 18x^3 + 20 \)[/tex] (Degree 5)
This ordering is based on the descending degrees of the polynomials.
1. Polynomial: [tex]\( 4x^2 + 5x^8 - 17x^3 - 5x^7 \)[/tex]
- Highest degree: [tex]\( x^8 \)[/tex]
- Degree: 8
2. Polynomial: [tex]\( 23 + x^9 \)[/tex]
- Highest degree: [tex]\( x^9 \)[/tex]
- Degree: 9
3. Polynomial: [tex]\( 3x^5 - 7x^4 + 18x^3 + 20 \)[/tex]
- Highest degree: [tex]\( x^5 \)[/tex]
- Degree: 5
4. Polynomial: [tex]\( 16x^5 - 4x^4 + 23x^3 - 21 + x^6 \)[/tex]
- Highest degree: [tex]\( x^6 \)[/tex]
- Degree: 6
Having found the degrees, we can now list the polynomials from the greatest degree to the least:
1. [tex]\( 23 + x^9 \)[/tex] (Degree 9)
2. [tex]\( 4x^2 + 5x^8 - 17x^3 - 5x^7 \)[/tex] (Degree 8)
3. [tex]\( 16x^5 - 4x^4 + 23x^3 - 21 + x^6 \)[/tex] (Degree 6)
4. [tex]\( 3x^5 - 7x^4 + 18x^3 + 20 \)[/tex] (Degree 5)
This ordering is based on the descending degrees of the polynomials.
