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The Fundamental Theorem of Algebra is a key concept in understanding polynomial functions. It states that every non-zero, single-variable polynomial of degree \( n \) has exactly \( n \) roots (also called zeros or solutions). These roots can be real or complex numbers. The theorem assures us that a polynomial will have a set number of roots equal to its degree, so a polynomial of degree 5 will have five roots.

This does not necessarily mean that all solutions are distinct; some roots can repeat. For example, if a polynomial has a root of \(-4\) repeated three times, \(-4\) is a root of multiplicity 3. This concept helps in constructing polynomials by knowing exactly how many roots must be considered, ensuring all possible solutions are accounted for.

The degree of a polynomial is the highest power of the variable in the polynomial expression. It plays a crucial role in determining the behavior and shape of the polynomial function. In our exercise, the degree is 5, indicating that our polynomial should represent a fifth-order function.

A high degree generally indicates a more complex graph, with possible twists and turns, but the degree also strictly determines the number of roots that the polynomial can have, according to the Fundamental Theorem of Algebra.

A polynomial degree also impacts the end behavior of the graph. For an odd-degree polynomial such as degree 5, the ends of the graph will move in opposite directions. This sets fundamental expectations for how the graph will behave as it extends towards positive and negative infinity.

Zeros of a polynomial, also known as roots or solutions, are the values of \( x \) that make the polynomial equal to zero. In simpler terms, these are the points where the graph of the polynomial crosses the x-axis. For the polynomial in our exercise, the specified zeros are \(0\) and \(-4\).

To find a polynomial with these zeros, we can use the factor method. A zero \( r \) corresponds to a factor \((x - r) \) for the polynomial function. Thus, if we know zeros of a polynomial, we can build its factors.

In our exercise, we have zeros at \( x=0 \) and \( x=-4 \). By including additional repeated roots of \( x = -4 \) to create a polynomial degree of 5, our polynomial is constructed as \( x \times (x + 4)^4 \), which ensures that each root is counted in the multiplicity needed to reach the required degree.