What are all the possible rational zeros for #f(x)=x^5+2x^4+14x^3+26x^2+40x+80# and how do you find all zeros?
By the rational roots theorem, any rational zeros of
In addition note that all of the coefficients of
So the only possible rational zeros of
#-1, -2, -4, -5, -8, -10, -16, -20, -40, -80#
None of these work, so
Typically for a quintic or polynomial of higher degree, the zeros of
About the best you can do is find numerical approximations using a method such as Durand-Kerner to find:
#x_1 ~~ -1.92974#
#x_(2,3) ~~ -0.135287+-3.0955i#
#x_(4,5) ~~ 0.100156+-2.07561i#
For more details on using Durand-Kerner to find such zeros of a quintic, see https://socratic.org/s/axney67c
Here's an example C++ program that implements the Durand-Kerner algorithm for our current example...