College

**Quiz: What Is a Polynomial?**

**Question 1 of 10:**

Which of the following shows the polynomial below written in descending order?

\[ 4x^2 - x + 8x^6 + 3 + 2x^{10} \]

A. \[ 2x^{10} + 8x^6 + 4x^2 - x + 3 \]

B. \[ 8x^6 + 4x^2 + 3 + 2x^{10} - x \]

C. \[ 2x^{10} + 4x^2 - x + 3 + 8x^6 \]

Answer :

Let's put the given polynomial [tex]\(4x^2 - x + 8x^6 + 3 + 2x^{10}\)[/tex] in descending order of the exponents of [tex]\(x\)[/tex].

First, identify the terms and their respective exponents:

- The term [tex]\(4x^2\)[/tex] has an exponent of 2.
- The term [tex]\(-x\)[/tex] can be interpreted as [tex]\(-1x^1\)[/tex], with an exponent of 1.
- The term [tex]\(8x^6\)[/tex] has an exponent of 6.
- The term [tex]\(3\)[/tex] can be interpreted as [tex]\(3x^0\)[/tex], with an exponent of 0.
- The term [tex]\(2x^{10}\)[/tex] has an exponent of 10.

To write the polynomial in descending order, we organize the terms by their exponents, starting with the highest:

1. The term with the highest exponent is [tex]\(2x^{10}\)[/tex].
2. The next highest exponent is 8 from the term [tex]\(8x^6\)[/tex].
3. The next is 4 from the term [tex]\(4x^2\)[/tex].
4. The next is -1 from the term [tex]\(-x\)[/tex], where [tex]\(x\)[/tex] has an exponent of 1.
5. The constant term [tex]\(3\)[/tex], where the exponent of [tex]\(x\)[/tex] is 0.

So in descending order, the polynomial should be organized as:

[tex]\[2x^{10} + 8x^6 + 4x^2 - x + 3\][/tex]

Now, let's check the options provided:

A. [tex]\(2x^{10} + 8x^6 + 4x^2 - x + 3\)[/tex]

B. [tex]\(8x^6 + 4x^2 + 3 + 2x^{10} - x\)[/tex]

C. [tex]\(2x^{10} + 4x^2 - x + 3 + 8x^6\)[/tex]

Comparing each option with our organized polynomial:

- Option A matches exactly with [tex]\(2x^{10} + 8x^6 + 4x^2 - x + 3\)[/tex].
- Option B starts with [tex]\(8x^6\)[/tex] rather than [tex]\(2x^{10}\)[/tex].
- Option C does not maintain the correct descending order.

Therefore, the correct answer is:

A. [tex]\(2x^{10} + 8x^6 + 4x^2 - x + 3\)[/tex]