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     Sketch a graph, , that has the number and types of roots described for each situation below.

  1. real roots

    Since the degree is an odd number, it will start and end in different directions.
    A real root occurs when the polynomial intersects the -axis, so this graph should intersect the -axis times.

  2. real and complex roots

    This polynomial has roots, but only -intercepts.

    A possible graph would be a polynomial with -intercepts plus another bend.

  3. complex roots

    The degree of this polynomial is even, so it starts and ends in the same direction.

  4. complex and real roots

    See part (a), (b), and (c).