For each polynomial function: (a) List each real zero and...

For each polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of $|x|$. $$ f(x)=4 x\left(x^{2}-3\right) $$

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the maximum number of turning points on the graph.
(d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of $|x|$.
$$
f(x)=4 x\left(x^{2}-3\right)
$$

Key Concepts

Polynomial Function

A polynomial function is an algebraic expression consisting of variables raised to non-negative integer powers and multiplied by coefficients. These functions are continuous and differentiable everywhere, and their graphs are smooth curves without breaks or sharp corners, making them a fundamental object of study in algebra and calculus.

Real Zeros and Their Multiplicities

Real zeros of a polynomial function are the x-values at which the function equals zero. The multiplicity of a zero is the number of times it appears as a root. A zero with a higher multiplicity means that the factor corresponding to that zero is repeated; this affects the shape of the graph at the intercept.

Graph Behavior at Intercepts

The way a polynomial graph interacts with the x-axis at its zeros depends on the multiplicity of the zeros. If the zero has an odd multiplicity, the graph typically crosses the x-axis, while if the multiplicity is even, it touches the x-axis and turns around at that point.

Turning Points

Turning points are the locations where a function changes direction, from increasing to decreasing or vice versa. The maximum number of turning points in a polynomial function is at most one less than its degree, which indicates the potential complexity of the graph's curvature.

End Behavior

The end behavior of a polynomial function describes the behavior of the graph as the input values approach positive or negative infinity. This behavior is determined by the leading term (the term with the highest power), which acts as a power function that dominates the polynomial for large absolute values of x.

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