MHB No. of real solutions in polynomial equation

[juantheron]

The no. of real solution of the equation $\displaystyle 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} = 0$

 

[mathbalarka]

Re: no. of real solution in polynomial equation

Spoiler

Analysis of the discriminant shows that the discriminant is 1/576 which is obviously > 0. So, either there are two pairs of complex conjugate roots or none at all.

So, check if the additional discriminant ($$8ac - 3b^2$$) is greater or smaller than 0. In this case, it is -1/3 < 0, and hence all the roots are distinct and real.

 

[Opalg]

Re: no. of real solution in polynomial equation

[sp]$ 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} = \frac1{4!}(x^4 + 4x^3 + 12x^2 + 24x + 24) = \frac1{4!}\bigl((x^2+2x+2)^2 + 4(x+2)^2 + 4\bigr) >0$ for all real $x$, so the equation has no real solutions.[/sp]