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Answer :
To determine which polynomial lists its terms in descending order (from the highest exponent to the lowest), we follow these steps:
1. Write down the exponent of the variable \( x \) for each term in the polynomial as it appears. (Recall that the constant term has an exponent of \(0\).)
2. Verify if these exponents are ordered from the largest to the smallest.
Let's analyze each option:
**Option A:**
The polynomial is
\[
10x^2 + 8x^3 + x^8 - 2 + 3x^6.
\]
The exponents in the order the terms appear are:
\[
2,\ 3,\ 8,\ 0,\ 6.
\]
This order is not descending since \(2 < 3 < 8\) and later \(0\) comes before \(6\).
**Option B:**
The polynomial is
\[
x^8 + 10x^2 + 8x^3 + 3x^6 - 2.
\]
The exponents in order are:
\[
8,\ 2,\ 3,\ 6,\ 0.
\]
This sequence is not entirely descending because after \(8\) the next exponent is \(2\), which should be followed by a number less than or equal to \(2\), but \(3\) and \(6\) come later.
**Option C:**
The polynomial is
\[
3x^6 + 10x^2 + x^8 + 8x^3 - 2.
\]
The exponents here are:
\[
6,\ 2,\ 8,\ 3,\ 0.
\]
Again, this does not follow the descending order since \(6\) is followed by \(2\) and then goes back up to \(8\).
**Option D:**
The polynomial is
\[
x^8 + 3x^6 + 8x^3 + 10x^2 - 2.
\]
The exponents in order are:
\[
8,\ 6,\ 3,\ 2,\ 0.
\]
This sequence is in descending order because each exponent is less than or equal to the one before it:
\[
8 > 6 > 3 > 2 > 0.
\]
Thus, the polynomial in Option D correctly lists the powers in descending order.
The final answer is: Option D.
1. Write down the exponent of the variable \( x \) for each term in the polynomial as it appears. (Recall that the constant term has an exponent of \(0\).)
2. Verify if these exponents are ordered from the largest to the smallest.
Let's analyze each option:
**Option A:**
The polynomial is
\[
10x^2 + 8x^3 + x^8 - 2 + 3x^6.
\]
The exponents in the order the terms appear are:
\[
2,\ 3,\ 8,\ 0,\ 6.
\]
This order is not descending since \(2 < 3 < 8\) and later \(0\) comes before \(6\).
**Option B:**
The polynomial is
\[
x^8 + 10x^2 + 8x^3 + 3x^6 - 2.
\]
The exponents in order are:
\[
8,\ 2,\ 3,\ 6,\ 0.
\]
This sequence is not entirely descending because after \(8\) the next exponent is \(2\), which should be followed by a number less than or equal to \(2\), but \(3\) and \(6\) come later.
**Option C:**
The polynomial is
\[
3x^6 + 10x^2 + x^8 + 8x^3 - 2.
\]
The exponents here are:
\[
6,\ 2,\ 8,\ 3,\ 0.
\]
Again, this does not follow the descending order since \(6\) is followed by \(2\) and then goes back up to \(8\).
**Option D:**
The polynomial is
\[
x^8 + 3x^6 + 8x^3 + 10x^2 - 2.
\]
The exponents in order are:
\[
8,\ 6,\ 3,\ 2,\ 0.
\]
This sequence is in descending order because each exponent is less than or equal to the one before it:
\[
8 > 6 > 3 > 2 > 0.
\]
Thus, the polynomial in Option D correctly lists the powers in descending order.
The final answer is: Option D.
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