Example	1: Determine	which	functions	are	polynomials.	State	the	degree	and	the	leading	coefficient	of
each	polynomial	function.
a) g(x)= sin x

To determine if a function is a polynomial, we check if it can be expressed in the form:

\[
f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
\]

where \(a_n, a_{n-1}, \dots, a_0\) are constants, \(n\) is a non-negative integer, and \(a_n \neq 0\).

### Function: \( g(x) = \sin x \)

- **Is it a polynomial?**
  - No, \(g(x) = \sin x\) is not a polynomial. The sine function is a trigonometric function, not a polynomial.

- **Degree:**
  - Since it's not a polynomial, it does not have a degree.

- **Leading Coefficient:**
  - Similarly, there is no leading coefficient because it's not a polynomial.

### Summary:
- **Function:** \( g(x) = \sin x \)
- **Polynomial:** No
- **Degree:** Not applicable
- **Leading Coefficient:** Not applicable

Would you like to go over more examples or have any questions? 

Here are five related questions for practice:
1. Determine if \( h(x) = x^3 - 2x^2 + x - 1 \) is a polynomial. State its degree and leading coefficient.
2. Is \( f(x) = \frac{1}{x} \) a polynomial? Explain why or why not.
3. What is the degree of the polynomial \( p(x) = 7x^5 - 3x^4 + 2 \)?
4. Identify the leading coefficient of \( q(x) = -4x^2 + x + 7 \).
5. Determine whether \( r(x) = e^x \) is a polynomial and state the reasoning.

**Tip:** A function is a polynomial only if all terms are of the form \(a_n x^n\) with \(n\) as a non-negative integer.

Explore whether g(x) = sin x is a polynomial function. Understand the definition of polynomials, their characteristics, and how to determine if a function like sine x qualifies as a polynomial. Suitable for Grades 9-12 mathematics students.

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