Which polynomial lists the powers in descending order?
A. $x^8+10 x^2+8 x^3+3 x^6-2$
B. $x^9+3 x^6+8 x^3+10 x^2-2$
C. $3 x^6+10 x^2+x^8+8 x^3-2$
D. $10 x^2+8 x^3+x^8-2+3 x^6$
Answer (1)
Examine each polynomial and identify the powers of x in each term.
Check if the powers are arranged from highest to lowest.
Option B: x 9 + 3 x 6 + 8 x 3 + 10 x 2 − 2 has powers 9, 6, 3, 2, 0, which are in descending order.
The polynomial with powers in descending order is B .
Explanation
Understanding the Problem We need to identify the polynomial that lists the powers of x in descending order. This means the exponent of x should decrease from left to right. Let's examine each option.
Analyzing Option A Option A: x 8 + 10 x 2 + 8 x 3 + 3 x 6 − 2 . The powers are 8, 2, 3, 6, 0. This is not in descending order.
Analyzing Option B Option B: x 9 + 3 x 6 + 8 x 3 + 10 x 2 − 2 . The powers are 9, 6, 3, 2, 0. This is in descending order.
Analyzing Option C Option C: 3 x 6 + 10 x 2 + x 8 + 8 x 3 − 2 . The powers are 6, 2, 8, 3, 0. This is not in descending order.
Analyzing Option D Option D: 10 x 2 + 8 x 3 + x 8 − 2 + 3 x 6 . The powers are 2, 3, 8, 0, 6. This is not in descending order.
Conclusion Therefore, the polynomial with powers in descending order is option B.
Examples
Understanding the order of polynomials is crucial in various fields, such as computer science for algorithm analysis and in physics for describing complex systems. For example, when analyzing the efficiency of a sorting algorithm, we might express the time complexity as a polynomial function of the input size, like n 2 + 3 n + 1 . Arranging this polynomial in descending order helps us quickly identify the dominant term ( n 2 ), which determines the algorithm's behavior for large inputs. This concept extends to modeling physical phenomena, where polynomial approximations are used to simplify complex equations, and the order of terms indicates their relative importance.
