Find a polynomial function that has the given zeros. 0,1,9

To find a polynomial function that has the given zeros \( 0, 1, \) and \( 9 \), we use the fact that if a polynomial has zeros \( a, b, c \), then the polynomial can be written in factored form as:

\[
f(x) = a(x - r_1)(x - r_2)(x - r_3)
\]
where \( r_1, r_2, r_3 \) are the zeros, and \( a \) is the leading coefficient (which we can take as 1 if not otherwise specified).

Given the zeros \( 0, 1, 9 \), the polynomial will be:

\[
f(x) = (x - 0)(x - 1)(x - 9)
\]
Simplify the expression:

\[
f(x) = x(x - 1)(x - 9)
\]
Now, expand it:

\[
f(x) = x \left[ (x - 1)(x - 9) \right]
\]
First, expand \( (x - 1)(x - 9) \):

\[
(x - 1)(x - 9) = x^2 - 9x - x + 9 = x^2 - 10x + 9
\]

Now multiply by \( x \):

\[
f(x) = x(x^2 - 10x + 9) = x^3 - 10x^2 + 9x
\]

Thus, the polynomial function is:

\[
f(x) = x^3 - 10x^2 + 9x
\]

### Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you verify that 0, 1, and 9 are indeed the zeros of the polynomial?
2. What happens to the polynomial if one of the zeros is repeated, for example, \( 0, 1, 1, 9 \)?
3. Can you express the polynomial in terms of its factored form if we add a leading coefficient?
4. How would the polynomial change if the zeros were complex numbers, such as \( 0, 1, i \)?
5. What would the graph of this cubic polynomial look like, and how do the zeros affect its shape?

**Tip:** The degree of the polynomial equals the number of its zeros (including repeated ones), which can help in identifying the type of polynomial you're dealing with.

Find a polynomial function that has the given zeros. 0, 1, 9

Discover how to find a polynomial function with the zeros 0, 1, and 9, using polynomial factorization and expansion. Learn the steps to convert a factored polynomial into its expanded form, explained with detailed calculations. Suitable for students in Grades 9-11.

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