Determine the equation of the 5th degree polynomial that touches the x-axis at (-3,0), crosses the y-axis at (0,-4.5) and crosses the x-axis at (1,0)

To determine the equation of a 5th degree polynomial that meets the given conditions, we need to follow a systematic approach. Let's denote the polynomial by \( P(x) \). 

### Conditions given:
1. The polynomial touches the x-axis at \((-3, 0)\). This means \(x = -3\) is a root with at least multiplicity 2.
2. The polynomial crosses the y-axis at \((0, -4.5)\). This means \( P(0) = -4.5 \).
3. The polynomial crosses the x-axis at \((1, 0)\). This means \(x = 1\) is a simple root.

### Step-by-step solution:

1. **Identify the roots and their multiplicities:**
   - Since the polynomial touches the x-axis at \(x = -3\), \((-3)\) must be a root with multiplicity at least 2.
   - Since the polynomial crosses the x-axis at \(x = 1\), \(1\) must be a simple root.

   Therefore, the polynomial can be written in the form:
   \[
   P(x) = a(x + 3)^2(x - 1)Q(x)
   \]
   where \(Q(x)\) is a polynomial of degree 2 because we need a polynomial of degree 5 in total.

2. **Determine \(Q(x)\):**
   Since we need a 5th degree polynomial and we already have 3 factors, \(Q(x)\) must be a quadratic polynomial. Assume:
   \[
   Q(x) = (x - r_1)(x - r_2)
   \]
   where \(r_1\) and \(r_2\) are the remaining roots.

3. **Use the y-intercept condition:**
   The polynomial crosses the y-axis at \((0, -4.5)\). Thus,
   \[
   P(0) = a(0 + 3)^2(0 - 1)(0 - r_1)(0 - r_2) = -4.5
   \]
   Simplifying, we get:
   \[
   9a(-1)(-r_1)(-r_2) = -4.5
   \]
   \[
   9a r_1 r_2 = -4.5
   \]
   \[
   a r_1 r_2 = -\frac{4.5}{9}
   \]
   \[
   a r_1 r_2 = -\frac{1}{2}
   \]

4. **Determine the unknowns:**
   Since we need additional information to determine \(r_1\) and \(r_2\), let's proceed assuming specific roots to solve the problem, or express them symbolically.

   Suppose \(r_1\) and \(r_2\) are such that \( r_1 = r \) and \( r_2 = s \). Then we have:
   \[
   P(x) = a(x + 3)^2(x - 1)(x - r)(x - s)
   \]
   with
   \[
   a \cdot r \cdot s = -\frac{1}{2}
   \]

5. **Solve for the roots \(r\) and \(s\):**
   Without additional constraints, assume simplest case:
   - Let \( r = 2 \), then \(s\) would be determined by:
   \[
   a \cdot 2 \cdot s = -\frac{1}{2}
   \]
   \[
   a \cdot s = -\frac{1}{4}
   \]
   If we assume \( a = 1 \):
   \[
   s = -\frac{1}{4}
   \]

6. **Construct the polynomial:**
   \[
   P(x) = (x + 3)^2(x - 1)(x - 2)\left(x + \frac{1}{4}\right)
   \]
   If we include a constant \(a\) to adjust for scaling:
   \[
   P(x) = a(x + 3)^2(x - 1)(x - 2)(4x + 1)
   \]
   Substitute \(P(0) = -4.5\) to find \(a\):
   \[
   -4.5 = a(3)^2(-1)(-2)(1) \Rightarrow -4.5 = a(9)(2)
   \]
   \[
   -4.5 = 18a \Rightarrow a = -\frac{4.5}{18} = -\frac{1}{4}
   \]

7. **Final polynomial:**
   \[
   P(x) = -\frac{1}{4} (x + 3)^2(x - 1)(x - 2)(4x + 1)
   \]

This is the polynomial of degree 5 that satisfies the given conditions.

Do you want more details or have any questions?

### Relative questions:
1. What is a polynomial's degree and why is it important?
2. What is the difference between a root's multiplicity and a simple root?
3. How do you determine a polynomial equation from its roots?
4. How can you use the y-intercept to find a polynomial's leading coefficient?
5. What are the implications of a polynomial touching the x-axis at a point?
6. How can we verify that a given polynomial meets specific conditions?
7. What role does the constant coefficient play in the polynomial equation?
8. How do you graph a polynomial to visualize its roots and behavior?

### Tip:
To understand the behavior of a polynomial function, always analyze its roots and their multiplicities, as well as how it intersects the axes.

Learn how to determine the equation of a 5th degree polynomial that touches the x-axis at (-3, 0), crosses the y-axis at (0, -4.5), and crosses the x-axis at (1, 0). This problem involves understanding polynomials, roots, and intercepts, suitable for advanced high school and college students.

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