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Polynomial zeros are the values of \(x\) that make the polynomial equal to zero. These are also known as the roots of the polynomial. For example, if \(x = -5, 0,\) and \(1\) are zeros of a polynomial, this means that substituting these values into the polynomial function results in the function equaling zero.

In the exercise, the zeros \(x = -5, 0,\) and \(1\) were provided. Each zero corresponds to a factor of the polynomial, highlighting the essential link between zeros and factors. Understanding how to determine these zeros is crucial for constructing polynomial functions. When you "solve" a polynomial, you are often seeking its zeros.

The degree of a polynomial is the highest power of \(x\) in the polynomial. It is a critical indicator of the polynomial's behavior and characteristics. For instance, a polynomial with a degree of 3 will have an \(x^3\) term as the highest power. This means it's a cubic polynomial.

The degree influences the shape and the number of zeros a polynomial function might have. In this exercise, we were tasked with finding a cubic polynomial (degree 3), which naturally accommodates the three zeros given: \(-5, 0,\) and \(1\). The degree also helps in predicting the number of possible turning points and the end behavior of the polynomial's graph.

Factorization involves expressing a polynomial as a product of its factors. This technique is particularly important because it makes it easier to find the polynomial's zeros. In our example, the zeros \(-5, 0,\) and \(1\) give rise to factors \((x + 5), (x - 0),\) and \((x - 1)\).

To form the polynomial, multiply these factors: \((x+5)(x-0)(x-1)\). By expanding the expression, you get \(x^3 + 4x^2 - 5x\), the polynomial that has the given zeros and meets the degree requirement.

Factorization not only aids in forming polynomials but also in breaking down complex polynomials into simpler, manageable components. It is a vital skill for both constructing and solving polynomial equations.