https://file.aiquickdraw.com/mathbot/file/1729085240464htkasvfw.jpg

The problem shown in the image states:

"The function \( f(x) = x(x - 2x^2) \) is a polynomial function of the ____."

The missing word refers to the degree of the polynomial.

### Let's solve the problem step by step:

1. Start by expanding the given function:
   \[
   f(x) = x(x - 2x^2) = x^2 - 2x^3
   \]
   So, the function is \( f(x) = -2x^3 + x^2 \).

2. Now, the degree of a polynomial is the highest power of \( x \). In this case, the highest power of \( x \) is 3 (from the term \( -2x^3 \)).

Thus, the polynomial function is of **third** degree.

So, the answer is (c) **third**.

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Would you like more details on polynomial degrees or have any other questions?

Here are some related questions for further understanding:
1. What defines the degree of a polynomial?
2. How do you determine the leading coefficient of a polynomial?
3. What is the significance of the degree in graphing polynomials?
4. Can a polynomial have multiple terms with the same degree?
5. What is the degree of a constant function?

**Tip**: Always expand and simplify polynomials before determining their degree.

The function f(x) = x(x - 2x^2) is a polynomial function of the ____.

Learn how to determine the degree of the polynomial function f(x) = x(x - 2x^2). This problem demonstrates key algebraic concepts, including expanding and simplifying polynomials and finding the highest power of x. Suitable for high school students.

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