How do you find all the zeros of #f(x)=(6x^7+4x^2+6)(x^4+5x-6)# with its multiplicities?
1 Answer
Find zeros of
Explanation:
#f(x) = (6x^7+4x^2+6)(x^4+5x-6)#
The zeros of
Zeros of
Notice that the sum of the coefficients is
Hence
#x^4+5x-6 = (x-1)(x^3+x^2+x+6)#
By the rational root theorem, the possible rational zeros of
#x^3+x^2+x+6 = (x+2)(x^2-x+3)#
The remaining quadratic has Complex zeros given by the quadratic formula:
#x = (1+-sqrt(-11))/2 = color(blue)(1/2+-sqrt(11)/2i)#
Zeros of
This septic has no simple factorisation and no rational zeros.
About the best you can do is use a numerical algorithm to find approximations for the
The Durand-Kerner method is straightforward to code and gives all of the approximate zeros at once.
#x ~~ -1.08646#
#x ~~ -0.539746+-0.832444i#
#x ~~ 0.15543+-0.896241i#
#x ~~ 0.927548+-0.519443i#
