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?

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.