To sketch a polynomial $f(x)$ with the given characteristics, let’s analyze each condition:

1. Polynomial Degree

• The polynomial has a degree of 4, which means the leading term will be $x^4$, making it a quartic polynomial.

2. Roots

• The polynomial has roots at:
  • $x = -4$ (multiplicity 1, so it crosses the x-axis at this root).
  • $x = -2$ (multiplicity 1, so it also crosses the x-axis here).
  • $x = 3$ (multiplicity 2, so it touches the x-axis and does not cross it at this root).

3. End Behavior

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

Since the end behavior of the polynomial shows that $f(x)$ approaches $-\infty$ on both ends, the leading coefficient must be negative. Thus, the polynomial will have the form: $f(x) = -a(x + 4)(x + 2)(x - 3)^2$ where $a > 0$ (to ensure the correct end behavior).

Steps to Sketch

1. Plot the roots on the x-axis at $x = -4$, $x = -2$, and $x = 3$.
2. Behavior at each root:
   • At $x = -4$ and $x = -2$: the function crosses the x-axis.
   • At $x = 3$: the function touches the x-axis and turns back, due to the multiplicity of 2.
3. End Behavior: The graph heads down to $-\infty$ as $x$ approaches both $-\infty$ and $+\infty$.

Would you like a specific graph of this polynomial, or further explanation on any of the steps?

Related Questions:

1. How does the multiplicity of a root affect the shape of the graph near that root?
2. Why does a negative leading coefficient in a quartic polynomial result in both ends going to $-\infty$?
3. What happens if we change the multiplicity of the root at $x = 3$ to 1?
4. How would the graph look if the leading coefficient were positive?
5. Can a quartic polynomial have end behavior where one end goes to $+\infty$ and the other to $-\infty$?

Tip:

When sketching polynomials, always check the end behavior first, as it gives you a good idea of the overall direction of the graph on both sides.