CPhill: How did Alan know that this polynomial had no real roots? Or did he do the calculation to find out?. Thanks.
+129840 By Descartes Rule of Signs....since f(x) has two sign changes, ..this polynomial either has 2 positive real roots, or no positive real roots
Look at the graph, here : https://www.desmos.com/calculator/coquksvvjl
Note that the graph never crosses the x axis at any point. Thus......there are no positive [ or negative] real roots....!!!!!
Here's an explanation of Descartes Rule of Signs......http://www.purplemath.com/modules/drofsign.htm
+33653 Another way of seeing there are no real roots is to rewrite the equation as:
x^4 + 3x^2 + 5 = x
The left-hand side is always greater than or equal to 5, whatever the value of x.
If x is less than 5 the right-hand side is obviously less than 5 and hence less than the left-hand side.
If x is greater than or equal to 5 the x^4 on the left-hand side, on it's own, is greater than the x on the right-hand side.
Hence there are no real values of x that satisfy the equation (i,e. there are no real roots).
