To determine which polynomial lists the powers of [tex]\( x \)[/tex] in descending order, let's review the provided polynomials and ensure that the exponents of [tex]\( x \)[/tex] are ordered from the highest to the lowest.
Let's analyze each option:
Option A:
[tex]\[ 4x^5 - 2x^2 - x^3 + 3x^4 + 1 \][/tex]
Here, the powers of [tex]\( x \)[/tex] are not in descending order. Specifically, [tex]\( x^2 \)[/tex] and [tex]\( x^3 \)[/tex] are placed incorrectly.
Option B:
[tex]\[ 3x^4 - x^3 + 4x^5 - 2x^2 + 1 \][/tex]
In this polynomial, [tex]\( 4x^5 \)[/tex] appears after [tex]\( 3x^4 \)[/tex] and [tex]\( -x^3 \)[/tex], so the order is not descending.
Option C:
[tex]\[ 1 - 2x^2 - x^3 + 4x^5 + 3x^4 \][/tex]
Again, the polynomial starts with the constant term and the powers of [tex]\( x \)[/tex] are not in the correct descending order.
Option D:
[tex]\[ 4x^5 + 3x^4 - x^3 - 2x^2 + 1 \][/tex]
In this polynomial:
- [tex]\( 4x^5 \)[/tex] (highest power)
- [tex]\( 3x^4 \)[/tex]
- [tex]\( -x^3 \)[/tex]
- [tex]\( -2x^2 \)[/tex]
- [tex]\( 1 \)[/tex] (constant term)
The polynomial lists the powers in descending order from [tex]\( x^5 \)[/tex] to [tex]\( x^0 \)[/tex].
Therefore, the correct polynomial that lists the powers of [tex]\( x \)[/tex] in descending order is:
D. [tex]\( 4x^5 + 3x^4 - x^3 - 2x^2 + 1 \)[/tex]
