Final answer:
The number of real solutions of the equation x^5 - 5x = p depends on the value of p. The exact real solutions can be found using numerical methods or graphical analysis, which involves sketching the graph of the equation and determining the intersections.
Step-by-step explanation:
The question asks us to find the number of real solutions of the equation x5 − 5x = p, where p is a real number. Analyzing the equation, we see that it is a quintic polynomial, which, in general, can have complex solutions. However, Descartes' Rule of Signs can help us predict the number of positive or negative real roots based on the number of sign changes in the polynomial.
Here, without specific values of p, we cannot provide a definite count of real solutions because the number of real solutions can vary based on the value of p. To find the exact real solutions, one would usually resort to numerical methods or graphical analysis, such as sketching the graph of f(x) = x5 − 5x and observing the intersections with the horizontal line y = p.