 Sketch the graph of a polynomial with a negative leading coefficient given
zeros -2 and 3 with multiplicity of 1 and 0 with multiplicity of 2

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.

Sketch the graph of a polynomial with a negative leading coefficient given zeros -2 and 3 with multiplicity of 1 and 0 with multiplicity of 2.

Learn how to sketch a polynomial graph with a negative leading coefficient, with zeros at -2 and 3 with multiplicity 1, and 0 with multiplicity 2. This solution covers polynomial characteristics, end behavior, and impacts of zero multiplicities, ideal for Grades 10-12.

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