Which of the following shows the polynomial below written in descending order?
[tex]3 x^3+9 x^7-x+4 x^{12}[/tex]
A. [tex]9 x^7+4 x^{12}+3 x^3-x[/tex]
B. [tex]4 x^{12}+3 x^3-x+9 x^7[/tex]
C. [tex]4 x^{12}+9 x^7+3 x^3-x[/tex]
D. [tex]3 x^3+4 x^{12}+9 x^7-x[/tex]
Answer (1)
Identify the exponents of each term in the polynomial.
Arrange the terms in decreasing order of their exponents.
Write the polynomial with the terms in the identified order.
The polynomial in descending order is 4 x 12 + 9 x 7 + 3 x 3 − x .
Explanation
Understanding the Problem We are given the polynomial 3 x 3 + 9 x 7 − x + 4 x 12 and asked to rewrite it in descending order. Descending order means arranging the terms from the highest power of x to the lowest power of x .
Identifying Exponents Let's identify the exponent of each term:
3 x 3 has an exponent of 3.
9 x 7 has an exponent of 7.
− x has an exponent of 1.
4 x 12 has an exponent of 12.
Arranging Terms Now, we arrange the terms in decreasing order of their exponents. The order is 4 x 12 , 9 x 7 , 3 x 3 , − x .
Final Answer Therefore, the polynomial in descending order is 4 x 12 + 9 x 7 + 3 x 3 − x . Comparing this with the given options, we see that option C matches our result.
Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a ball, the growth of a population, or the behavior of electrical circuits. Arranging polynomials in descending order makes it easier to analyze their behavior and perform calculations, such as finding roots or evaluating the polynomial for a specific value of x. For example, if we have a polynomial representing the height of a projectile over time, writing it in descending order helps us quickly identify the dominant term, which often corresponds to the initial velocity or gravitational force.
