Expert: Steve Holleran - 8/8/2008

Question
Hi. I'm reviewing out of my Calculus I book this summer because I plan to take Calculus II next semester. In the chapter discussing critical points a question reads: "How many real roots does the equation x^5+x+1=0 have? How do you know? [Hint: How many critical points does this function have?]" The hint actually made it more confusing for me, because it seems to me that the critical point is irrelevant to finding the answer. Can you help?

Answer
Hi Tim,I think what they are getting at is that when you take the derivative,

                     x^5 + x + 7 =0

   so                5x^4 + 1 = 0  to find critical points, there are none, so that means the curve doesn't have any local max/min turning points, so therefore , it can only cross the x-axis once.

Remember that a 5th degree polynomial can have 5, 3 or 1 real root (Descartes Rule of Signs),  so if it has no turning points, that means it only has 1 rea zero.

Hope this helps
Steve