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To find a polynomial function of degree 3 with given zeros \( -3 \), \( 4 \), and \( 6 \), and a leading coefficient of 1, we can use the following steps:

### Step-by-step process:
1. **Form the factors** based on the zeros. The polynomial can be written as a product of factors corresponding to its zeros:
   \[
   f(x) = (x - (-3))(x - 4)(x - 6)
   \]
   This simplifies to:
   \[
   f(x) = (x + 3)(x - 4)(x - 6)
   \]

2. **Expand the product** of these factors:
   First, multiply the first two factors:
   \[
   (x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12
   \]
   Now, multiply this result by the remaining factor \( (x - 6) \):
   \[
   (x^2 - x - 12)(x - 6) = x^3 - 6x^2 - x^2 + 6x - 12x + 72
   \]
   Simplify the expression:
   \[
   f(x) = x^3 - 7x^2 + 6x + 72
   \]

Thus, the polynomial function is:
\[
f(x) = x^3 - 7x^2 + 6x + 72
\]

Would you like more details on the expansion process or have any questions?

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Here are 5 related questions you can explore:
1. How do you find a polynomial function if some of the zeros are repeated?
2. What happens if a zero is a complex number?
3. How would the polynomial change if the leading coefficient was not 1?
4. Can you explain the relationship between the degree of a polynomial and the number of zeros it can have?
5. How do you factor a cubic polynomial if the zeros are not given?

**Tip:** To check your work, remember that the sum of the zeros should be equal to the negative of the coefficient of the \(x^2\)-term, divided by the leading coefficient.

Find a polynomial function of degree 3 with the given numbers as zeros. Assume that the leading coefficient is 1. The zeros are: -3, 4, 6.

Learn how to find a cubic polynomial with the given zeros -3, 4, and 6. This problem covers key algebraic concepts such as factoring, the zero-product property, and polynomial expansion. Perfect for students in Grades 9-12.

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