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The problem asks whether the following functions are second-degree polynomials (trinomial du second degré), and to justify the answer.

### Function Analysis:

#### a) \( f(x) = (x - 1)^2 - x^2 \)

1. **Expand the expression**: 
   \[
   (x - 1)^2 = x^2 - 2x + 1
   \]
   So,
   \[
   f(x) = (x^2 - 2x + 1) - x^2
   \]
   Simplify the expression:
   \[
   f(x) = x^2 - 2x + 1 - x^2 = -2x + 1
   \]

   **Conclusion**: \( f(x) \) is a linear function (degree 1), not a second-degree polynomial (trinomial).

#### b) \( g(x) = x^4 + \frac{1}{x} - x(x^2 + x) \)

1. **Simplify the function**:
   \[
   g(x) = x^4 + \frac{1}{x} - (x \cdot x^2 + x \cdot x)
   \]
   \[
   g(x) = x^4 + \frac{1}{x} - (x^3 + x^2)
   \]
   \[
   g(x) = x^4 - x^3 - x^2 + \frac{1}{x}
   \]

   **Conclusion**: \( g(x) \) is not a second-degree polynomial since it contains terms of higher degree (like \(x^4\)) and a term with a negative degree (\(\frac{1}{x}\)).

### Final Answer:
- \( f(x) \) is a linear function, so it is not a second-degree trinomial.
- \( g(x) \) contains terms of various degrees (including fourth degree and a negative degree), so it is not a second-degree trinomial.

Would you like more details or have any questions? 

Here are five related questions you might find useful:
1. What are the characteristics of a second-degree polynomial?
2. How do you identify the degree of a polynomial function?
3. Can a second-degree polynomial have more than three terms?
4. How do the properties of exponents affect the classification of polynomial functions?
5. What is the general form of a second-degree trinomial?

**Tip:** Always look for the highest power of \(x\) in a polynomial to determine its degree.

Les fonctions suivantes sont-elles des fonctions trinômes du second degré ? Justifier. a) f(x) = (x - 1)^2 - x^2 b) g(x) = x^4 + 1/x - x(x^2 + x)

Analyze if given functions are second-degree polynomials by expanding and simplifying the expressions. Includes detailed solutions to determine the degree of polynomials and explanations for identifying algebraic functions. Suitable for high school students.

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