High School

Which polynomial lists the powers in descending order?

A. [tex]3x^6 + 10x^2 + x^8 + 8x^3 - 2[/tex]

B. [tex]10x^2 + 8x^3 + x^8 - 2 + 3x^6[/tex]

C. [tex]x^8 + 10x^2 + 8x^3 + 3x^6 - 2[/tex]

D. [tex]x^8 + 3x^6 + 8x^3 + 10x^2 - 2[/tex]

Answer :

To arrange the terms of a polynomial in descending order, we need to order the terms by the exponent on $x$ from the highest to the lowest. The constant term can be thought of as having an exponent of zero.

Let's examine option D:

$$
x^8 + 3x^6 + 8x^3 + 10x^2 - 2.
$$

In this polynomial, the exponents are:

- The first term has $x^8$ (exponent 8).
- The second term has $3x^6$ (exponent 6).
- The third term has $8x^3$ (exponent 3).
- The fourth term has $10x^2$ (exponent 2).
- The final term is $-2$, which corresponds to $x^0$ (exponent 0).

Since the exponents $8, 6, 3, 2, 0$ are in descending order, this polynomial is correctly written with the powers in descending order.

Thus, the correct answer is:

$$
\textbf{D. } x^8 + 3x^6 + 8x^3 + 10x^2 - 2.
$$