Question

Which of the following graphs represents the function f(x) = x4 − 2x3 − 9x2 + 2x + 8? graph with 3 real zeros, down on left, up on right graph with 3 real zeros, up on left, down on right graph with 4 real zeros, down on left, down on right graph with 4 real zeros, up on left, up on right

Answer 1

Answer:

4 real zeros, up on left, up on right

Step-by-step explanation:

The positive coefficient of x^4 tells you the graph will open upward (up on left, up on right). Descarte's rule of signs tells you there will be 0 or 2 positive real roots and 0 or 2 negative real roots. Based solely on end-behavior, the appropriate choice is the last one: ... up on left, up on right.

_____

More about the zeros

The sum of coefficients is zero, indicating x=1 is a root (zero). Hence there will be 2 positive real roots. Changing the signs of the coefficients of odd-degree terms also gives a sum of coefficients that is zero, indicating x=-1 is a root and that there are 2 negative real roots. At this point, you have enough information to factor the function completely, if you want. You also know there are 4 real roots.