For the polynomial $f(x) = 4 - x - 3x^2$, we will analyze the following properties:

1. Degree:

The degree of a polynomial is determined by the highest exponent of the variable $x$. Here, the term with the highest power of $x$ is $-3x^2$, which has an exponent of 2. Therefore, the degree of the polynomial is:

Degree = 2

2. Leading Term:

The leading term is the term with the highest degree. In this case, the leading term is:

Leading term = -3x^2

3. Leading Coefficient:

The leading coefficient is the coefficient of the leading term, which is the number in front of the highest-degree term. In this case, the leading coefficient is:

Leading coefficient = -3

4. Constant Term:

The constant term is the term without any variable $x$, which is just a number. In this polynomial, the constant term is:

Constant term = 4

5. End Behavior:

The end behavior of a polynomial is determined by the degree and the leading coefficient.

• Since the degree is even (2), the ends of the graph will go in the same direction.
• Since the leading coefficient is negative (-3), both ends will go down as $x$ approaches $+\infty$ and $-\infty$.

Thus, the end behavior is:

As $x \to \infty$, $f(x) \to -\infty$

As $x \to -\infty$, $f(x) \to -\infty$

Summary:

• Degree: 2
• Leading term: $-3x^2$
• Leading coefficient: $-3$
• Constant term: 4
• End behavior: Both ends go to $-\infty$

Would you like to dive deeper into any of these properties?
Here are some related questions you might find interesting:

1. How can the graph of this polynomial be sketching based on its degree and leading coefficient?
2. How do end behavior change when the degree is odd?
3. What happens to the graph of a polynomial if the constant term changes?
4. Can you explain how to factor a quadratic polynomial like this?
5. What are the possible real roots of this polynomial?

Tip: For polynomials, the degree helps determine the number of possible turning points on the graph, and the leading coefficient helps us understand the direction of the graph as $x \to \infty$ or $x \to -\infty$.