find a polynomial f(x) of degree 5 that has the f zeros 0,6(multiplicity 2), -5, -2

To find a polynomial \( f(x) \) of degree 5 with the given zeros, we follow these steps:

### Zeros and their corresponding factors:
- \( x = 0 \) corresponds to the factor \( x \).
- \( x = 6 \) has multiplicity 2, so it corresponds to the factor \( (x - 6)^2 \).
- \( x = -5 \) corresponds to the factor \( (x + 5) \).
- \( x = -2 \) corresponds to the factor \( (x + 2) \).

### Forming the polynomial:
The general form of the polynomial is:

\[
f(x) = C \cdot x \cdot (x - 6)^2 \cdot (x + 5) \cdot (x + 2)
\]

where \( C \) is a constant. If no specific conditions are given for the value of \( f(x) \) at some point (e.g., \( f(1) = 10 \)), we can assume \( C = 1 \).

Thus, the polynomial becomes:

\[
f(x) = x \cdot (x - 6)^2 \cdot (x + 5) \cdot (x + 2)
\]

### Expansion of the polynomial:
We can expand this expression step by step.

1. Expand \( (x - 6)^2 \):

\[
(x - 6)^2 = x^2 - 12x + 36
\]

2. Multiply by \( x \):

\[
x \cdot (x^2 - 12x + 36) = x^3 - 12x^2 + 36x
\]

3. Multiply by \( (x + 5) \):

\[
(x^3 - 12x^2 + 36x)(x + 5) = x^4 + 5x^3 - 12x^3 - 60x^2 + 36x^2 + 180x
= x^4 - 7x^3 - 24x^2 + 180x
\]

4. Finally, multiply by \( (x + 2) \):

\[
(x^4 - 7x^3 - 24x^2 + 180x)(x + 2) = x^5 + 2x^4 - 7x^4 - 14x^3 - 24x^3 - 48x^2 + 180x^2 + 360x
\]
\[
= x^5 - 5x^4 - 38x^3 + 132x^2 + 360x
\]

Thus, the expanded polynomial is:

\[
f(x) = x^5 - 5x^4 - 38x^3 + 132x^2 + 360x
\]

Let me know if you want further details or have any questions!

### 5 Related Questions:
1. What is the significance of the multiplicity of a root?
2. How do you find the degree of a polynomial given its zeros?
3. Can we determine the constant \( C \) if more information is provided?
4. How can we use synthetic division to verify a factor of a polynomial?
5. What are the applications of knowing the zeros of a polynomial?

### Tip:
When multiplying polynomials, it's helpful to work in smaller steps, expanding two factors at a time to avoid errors.

find a polynomial f(x) of degree 5 that has the f zeros 0, 6 (multiplicity 2), -5, -2

Explore the process of finding a degree 5 polynomial with given zeros, including 0, 6 (with multiplicity 2), -5, and -2. This detailed solution involves factoring, expanding polynomials, and applying key algebraic concepts like the Factor Theorem and Zero Product Property. Suitable for high school students (Grades 10-12).

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