Find a polynomial function​ f(x) of degree 3 with real coefficients that satisfies the following conditions.
Zero of 0 and zero of 4 having multiplicity​ 2; ​f(5​)equals=20

To find a polynomial function \( f(x) \) of degree 3 with the specified zeros and conditions, we start with the zeros:

1. A zero of \( 0 \) with multiplicity 2 means the factor \( x^2 \).
2. A zero of \( 4 \) with multiplicity 1 means the factor \( (x - 4) \).

Thus, the polynomial can be expressed as:

\[
f(x) = k \cdot x^2 (x - 4)
\]

where \( k \) is a constant to be determined.

Next, we expand the polynomial:

\[
f(x) = k \cdot x^2 (x - 4) = k \cdot (x^3 - 4x^2)
\]

Now we apply the condition \( f(5) = 20 \):

\[
f(5) = k \cdot (5^3 - 4 \cdot 5^2)
\]
\[
= k \cdot (125 - 100) = k \cdot 25
\]
Setting this equal to 20 gives us:

\[
k \cdot 25 = 20 \quad \Rightarrow \quad k = \frac{20}{25} = \frac{4}{5}
\]

Thus, the polynomial function is:

\[
f(x) = \frac{4}{5} \cdot (x^3 - 4x^2)
\]

We can rewrite it as:

\[
f(x) = \frac{4}{5}x^3 - \frac{16}{5}x^2
\]

### Summary
The polynomial function is:

\[
f(x) = \frac{4}{5}x^3 - \frac{16}{5}x^2
\]

### Would you like more details or have any questions? Here are some related questions you might consider:
1. What are the properties of polynomial functions of degree 3?
2. How do multiplicities of zeros affect the graph of a polynomial?
3. Can you find the derivative of this polynomial?
4. How would you determine the zeros of this polynomial?
5. What does it mean for a polynomial to have real coefficients? 

**Tip:** When analyzing polynomials, always check how the multiplicities of the zeros influence the shape of the graph.

Find a polynomial function f(x) of degree 3 with real coefficients that satisfies the following conditions. Zero of 0 and zero of 4 having multiplicity 2; f(5) = 20.

Discover how to construct a polynomial function of degree 3 with real coefficients, given specified zeros and multiplicities. This example guides you through identifying factors based on zero multiplicities and using function values to determine constants, suitable for high school math students.

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