To write the given polynomial [tex]\( 5x^3 - x + 9x^7 + 4 + 3x^{11} \)[/tex] in descending order, we need to arrange the terms such that the powers of [tex]\( x \)[/tex] are in decreasing order.
1. Identify the terms and their corresponding exponents:
- [tex]\( 3x^{11} \)[/tex] (exponent 11)
- [tex]\( 9x^7 \)[/tex] (exponent 7)
- [tex]\( 5x^3 \)[/tex] (exponent 3)
- [tex]\( -x \)[/tex] (exponent 1)
- [tex]\( 4 \)[/tex] (constant term which has exponent 0)
2. Sort these terms by their exponents in descending order:
- Highest exponent: [tex]\( 3x^{11} \)[/tex]
- Next: [tex]\( 9x^7 \)[/tex]
- Next: [tex]\( 5x^3 \)[/tex]
- Next: [tex]\( -x \)[/tex]
- Lowest exponent: [tex]\( 4 \)[/tex]
The polynomial written in descending order of the exponents will be:
[tex]\[ 3x^{11} + 9x^7 + 5x^3 - x + 4 \][/tex]
3. Now, match this ordered polynomial with the provided choices:
- A: [tex]\( 4 + 3x^{11} + 9x^7 + 5x^3 - x \)[/tex]
- B: [tex]\( 3x^{11} + 9x^7 - x + 4 + 5x^3 \)[/tex]
- C: [tex]\( 9x^7 + 5x^3 + 4 + 3x^{11} - x \)[/tex]
- D: [tex]\( 3x^{11} + 9x^7 + 5x^3 - x + 4 \)[/tex]
The correct choice that matches [tex]\( 3x^{11} + 9x^7 + 5x^3 - x + 4 \)[/tex] is:
D. [tex]\( 3x^{11} + 9x^7 + 5x^3 - x + 4 \)[/tex]
