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 resembles for large values of $|x| .$ $$f(x)=3(x-7)(x+3)^{2}$$

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 resembles for large values of $|x| .$
$$f(x)=3(x-7)(x+3)^{2}$$

Key Concepts

Polynomial Zeros and Multiplicity

Polynomial zeros are the x-values that make the polynomial equal to zero, and each zero has an associated multiplicity indicating how many times that factor is repeated. The multiplicity of a zero not only tells you how many times the factor appears in the factored form of the polynomial but also influences the local behavior of the graph at that point.

Graph Behavior at X-Intercepts

The behavior of the graph at an x-intercept is dictated by the multiplicity of the corresponding zero. If the multiplicity is odd, the graph will cross the x-axis at that zero; if it is even, the graph will only touch the x-axis and turn around at that point. This concept helps in predicting the shape of the polynomial graph near its zeros.

Turning Points

Turning points are the points on the graph of a polynomial where the function changes direction from increasing to decreasing or vice versa. For a polynomial of degree n, the maximum number of turning points is n-1, which provides an upper bound on the number of local extrema the polynomial can have.

End Behavior

The end behavior of a polynomial describes how the function behaves as x approaches positive or negative infinity, and it is determined by the leading term of the polynomial (the term with the highest power). This concept helps predict the overall direction of the graph for large absolute values of x, resembling a power function determined by the degree and sign of the leading coefficient.

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