In Exercises 30–31, find the zeros for each polynomial...

In Exercises 30–31, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero.
$$
f(x)=x^{3}-5 x^{2}-25 x+125
$$

Key Concepts

Polynomial Functions

A polynomial function is an expression made up of constants and variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Understanding the structure of polynomial functions is fundamental to studying their behavior and solving for their roots.

Zeros of a Polynomial

The zeros of a polynomial (also known as roots) are the values of the variable for which the function equals zero. Finding the zeros is equivalent to determining where the graph of the polynomial crosses or touches the x-axis.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a factor in the polynomial. A zero with a multiplicity greater than one indicates that its corresponding factor is repeated in the polynomial's factorization.

Graph Behavior at Zeros

The behavior of the graph at a zero depends on the multiplicity of that zero. If a zero has an odd multiplicity, the graph typically crosses the x-axis at that point; if it has an even multiplicity, the graph generally touches the x-axis and turns around, resulting in a bounce-off effect.

Polynomial Factorization Techniques

Techniques such as long division, synthetic division, and factoring by grouping are essential tools for breaking down a polynomial into simpler factors. Factoring assists in identifying the zeros and their multiplicities, which in turn provides insight into the overall graph of the function.

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