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In polynomials, a zero is a value for which the polynomial equals zero when substituted for the variable. Zeros can occur more than once for a single polynomial, and this is represented by the term "multiplicity."
To understand multiplicity better, consider the zeros of the given polynomial:

• A zero at \(-4\) with multiplicity 2 means the factor \((x + 4)\) appears twice in the polynomial.
• Similarly, a zero at \(5\) with multiplicity 3 means the factor \((x - 5)\) appears three times.

The concept of multiplicity helps to shape the polynomial's behavior. If a zero has a high multiplicity, the graph will "touch" the x-axis at this point but not cross it, creating a flat section on the curve. For zeros with an odd multiplicity, the graph will cross the x-axis. Understanding zeros and their multiplicities is crucial in constructing polynomials and predicting their graphs.

Polynomial expansion is the process of converting a factored form of a polynomial into a standard form where all the expressions are combined.
For this exercise, we start with the polynomial given in its factored state:

• \((x + 4)^2(x - 5)^3\)

The steps are as follows:

• First, expand the square: \((x + 4)^2 = x^2 + 8x + 16\).
• Next, expand the cube: \((x - 5)^3 = (x - 5)(x - 5)(x - 5) = x^3 - 15x^2 + 75x - 125\).
• Finally, multiply these two expanded expressions: \((x^2 + 8x + 16)(x^3 - 15x^2 + 75x - 125)\).

This expansion process involves distributing each term across the expressions and then combining like terms to achieve a polynomial in standard form. The expanded product yields a polynomial such as the one detailed in the solution: \[x^5 - 23x^4 + 148x^3 - 400x^2 + 600x - 320.\] This standard form shows the polynomial's degree and coefficients.

The degree of a polynomial is the highest power of the variable present. Finding the minimum degree for a polynomial involves ensuring that all the given zeros, along with their multiplicities, are accounted for with no extra factors.
In the exercise, zeros \(-4\) and \(5\) with respective multiplicities resulted in factors \((x + 4)^2\) and \((x - 5)^3\). The total degree becomes the sum of the multiplicities:

• 2 (for \(-4\))
• 3 (for \(5\))

Therefore, the sum of the multiplicities gives 2+3 = 5. Hence, the minimum degree of the polynomial is 5. This ensures the polynomial is neither unnecessarily complex nor missing required components. Minimum degree polynomials offer efficiency in calculations while fully representing the necessary characteristics of the function, adhering strictly to the zeros and their multiplicities.