(a) How many x -intercepts and how many local extrema does...

(a) How many $x$ -intercepts and how many local extrema does the polynomial $P(x)=x^{3}-4 x$ have? (b) How many $x$ -intercepts and how many local extrema does the polynomial $Q(x)=x^{3}+4 x$ have? (c) If $a>0,$ how many $Q(x)=x^{3}+4 x$ how many local extrema does each of the polynomials $P(x)=x^{3}-a x$ and $Q(x)=x^{3}+a x$ have? Explain your answer.

(a) How many $x$ -intercepts and how many local extrema
does the polynomial $P(x)=x^{3}-4 x$ have?
(b) How many $x$ -intercepts and how many local extrema
does the polynomial $Q(x)=x^{3}+4 x$ have?
(c) If $a>0,$ how many $Q(x)=x^{3}+4 x$ how many local
extrema does each of the polynomials $P(x)=x^{3}-a x$
and $Q(x)=x^{3}+a x$ have? Explain your answer.

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Key Concepts

X-intercepts

X-intercepts are the points where the graph of a polynomial function crosses the x-axis; these are the real solutions to the equation P(x)=0. The number of x-intercepts of a polynomial is determined by its degree and the nature of its roots. In the context of cubic functions, factorization often reveals the existence of one or three real roots, indicating the number of x-intercepts.

Local Extrema

Local extrema refer to the points on a graph where the function reaches a local maximum or minimum. For polynomial functions, these extrema are found by examining critical points, which are determined by setting the first derivative equal to zero. The number of local extrema in a cubic function can vary depending on the coefficients of the function, with derivatives having distinct real solutions indicating the presence of turning points.

Polynomial Functions

Polynomial functions, particularly cubic functions, are expressions where the highest power of x is three. Their end behavior is determined by the leading coefficient, and they can have up to three real roots and up to two turning points. Understanding these functions is crucial to analyzing their graphs and behavior across the domain.

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