To sketch the graph of this polynomial, let’s first analyze its main characteristics.

Key Points:

1. Zeros and Multiplicities:

   • The zeros are $x = -2$, $x = 3$, and $x = 0$.
   • $x = -2$ and $x = 3$ each have a multiplicity of 1, meaning the graph crosses the x-axis at these points.
   • $x = 0$ has a multiplicity of 2, so the graph touches but does not cross the x-axis here (a “bounce”).

2. Negative Leading Coefficient:

   • Since the leading coefficient is negative, the polynomial’s end behavior will point downward on both ends of the graph.

3. Polynomial Degree:

   • Summing the multiplicities of all zeros, we get $1 + 1 + 2 = 4$, so the polynomial is of degree 4.

General Form of the Polynomial

A polynomial with these characteristics could be written as: $f(x) = -a(x + 2)(x - 3)x^2$ where $a$ is a positive constant (the negative sign accounts for the negative leading coefficient).

Sketching the Graph:

1. End Behavior: Since this is a degree-4 polynomial with a negative leading coefficient, the graph will fall as $x \to \pm \infty$.
2. At $x = -2$: The graph crosses the x-axis.
3. At $x = 0$: The graph touches the x-axis and turns back, due to the multiplicity of 2.
4. At $x = 3$: The graph crosses the x-axis again.

Would you like further details on any part of this explanation?

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Here are five related questions:

1. How does the degree of a polynomial affect its end behavior?
2. What is the impact of a zero’s multiplicity on a graph’s shape at that point?
3. How do we determine the leading coefficient’s effect on the graph?
4. Can the shape of a polynomial graph vary if multiplicities change?
5. How do you find turning points in polynomial functions?

Tip: When sketching polynomials, focusing on zeros, multiplicities, and leading coefficient gives a quick view of the graph’s structure.