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To write a polynomial function of the least degree with given zeros, we can use the fact that each zero corresponds to a factor of the polynomial.

### Problem Analysis
The zeros given are: \(-1\), \(0\), \(1\), and \(-4\).

### Solution Steps

1. **Translate Zeros to Factors**:
   - For a zero at \( x = -1 \), the factor is \( (x + 1) \).
   - For a zero at \( x = 0 \), the factor is \( x \).
   - For a zero at \( x = 1 \), the factor is \( (x - 1) \).
   - For a zero at \( x = -4 \), the factor is \( (x + 4) \).

2. **Construct the Polynomial**:
   - Multiply the factors together to create the polynomial:
     \[
     g(x) = x(x + 1)(x - 1)(x + 4)
     \]

3. **Expand the Expression**:
   - First, simplify \((x + 1)(x - 1)\):
     \[
     (x + 1)(x - 1) = x^2 - 1
     \]
   - Next, simplify \((x)(x + 4)\):
     \[
     x(x + 4) = x^2 + 4x
     \]
   - Now, multiply these results together:
     \[
     g(x) = (x^2 + 4x)(x^2 - 1)
     \]

4. **Final Expansion**:
   - Expand \( (x^2 + 4x)(x^2 - 1) \):
     \[
     g(x) = x^2 \cdot x^2 + x^2 \cdot (-1) + 4x \cdot x^2 + 4x \cdot (-1)
     \]
     \[
     = x^4 - x^2 + 4x^3 - 4x
     \]

5. **Final Answer**:
   \[
   g(x) = x^4 + 4x^3 - x^2 - 4x
   \]

So, the polynomial function of the least degree with the specified zeros is:
\[
g(x) = x^4 + 4x^3 - x^2 - 4x
\]

Would you like further clarification or additional details on the steps?

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### Additional Questions to Explore:
1. What happens to the polynomial if a zero is repeated?
2. How does the degree of a polynomial relate to the number of its zeros?
3. What does it mean for a polynomial to have a "leading coefficient" of 1?
4. How can we verify that these are the only zeros of the polynomial?
5. Can this polynomial have any other factors without changing its degree?

### Tip
Always check that the number of factors matches the degree of the polynomial to ensure it has the minimal possible degree.

Write a polynomial function of least degree with zeros -1, 0, 1, and -4.

Learn how to construct a polynomial function of the least degree with zeros at -1, 0, 1, and -4. This solution includes the step-by-step expansion of the factors, creating a polynomial in standard form with a leading coefficient of 1. Suitable for students in Grades 10-12.

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