Find a polynomial function that has the given zeros. (There are many correct answers.)

0, 1, 5

To find a polynomial function with the given zeros \( 0 \), \( 1 \), and \( 5 \), we can write the polynomial as a product of factors corresponding to these zeros.

The factors will be of the form \( (x - \text{zero}) \). So, the factors for the given zeros are:

- \( x - 0 = x \)
- \( x - 1 \)
- \( x - 5 \)

Now, multiply these factors together to get the polynomial:

\[
f(x) = x(x - 1)(x - 5)
\]

Now expand this expression:

1. Multiply the first two factors:

\[
x(x - 1) = x^2 - x
\]

2. Now multiply by the third factor \( (x - 5) \):

\[
(x^2 - x)(x - 5) = x^3 - 5x^2 - x^2 + 5x = x^3 - 6x^2 + 5x
\]

Thus, one possible polynomial function is:

\[
f(x) = x^3 - 6x^2 + 5x
\]

### Final Answer:

\[
f(x) = x^3 - 6x^2 + 5x
\]

This is a polynomial with the given zeros \( 0 \), \( 1 \), and \( 5 \). There are infinitely many such polynomials because you can multiply this polynomial by any constant, and the resulting function will still have the same zeros.

Would you like a more detailed explanation of the process? Here are some related questions for further practice:

1. What happens to the polynomial if we multiply it by a constant, say 2?
2. How do you determine the degree of the polynomial from the number of zeros?
3. Can a polynomial with real coefficients have complex zeros?
4. How would you verify that these zeros satisfy the polynomial?
5. How do you find a polynomial if given complex or repeated zeros?

**Tip:** A polynomial of degree \( n \) has at most \( n \) zeros, including multiplicity.

Find a polynomial function that has the given zeros: 0, 1, 5

Learn how to find a polynomial function with zeros 0, 1, and 5 by factoring and expanding. This detailed step-by-step solution covers polynomial algebra, including factoring and the Fundamental Theorem of Algebra. Suitable for students in Grades 10-12.

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