How do you find all the zeros of f(x)=(6x^7+4x^2+6)(x^4+5x-6)f(x)=(6x7+4x2+6)(x4+5x6) with its multiplicities?

1 Answer
Jun 24, 2016

Find zeros of (x^4+5x-6)(x4+5x6) algebraically and approximate zeros of (6x^7+4x+6)(6x7+4x+6) numerically.

Explanation:

f(x) = (6x^7+4x^2+6)(x^4+5x-6)f(x)=(6x7+4x2+6)(x4+5x6)

The zeros of f(x)f(x) are the zeros of (6x^7+4x^2+6) = 2(3x^7+2x^2+3)(6x7+4x2+6)=2(3x7+2x2+3) and the zeros of (x^4+5x-6)(x4+5x6)

color(white)()
Zeros of bb (x^4+5x-6)

Notice that the sum of the coefficients is 0. That is: 1+5-6 = 0.

Hence x=color(blue)(1) is a zero and (x-1) a factor.

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 are: +-1, +-2, +-3. Note that color(blue)(-2) is a zero and (x+2) a factor:

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)

color(white)()
Zeros of bb (6x^7+4x^2+6 = 2(3x^7+2x^2+3))

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 1 Real and 6 non-Real Complex zeros.

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