How do you find all zeros of the function #f(x) = 3x^5 - x^2 + 2x + 18#?

1 Answer
Aug 6, 2016

Answer:

Use a numerical method to find approximations for the zeros:

#x_1 ~~ -1.35047#

#x_(2,3) ~~ -0.506174+-1.35526i#

#x_(4,5) ~~ 1.18141+-0.852685i#

Explanation:

#f(x) = 3x^5-x^2+2x+18#

By the rational root theorem, any rational zeros of #f(x)# are expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #18# and #q# a divisor of the coefficient #3# of the leading term.

That means that the only possible rational zeros are:

#+-1/3, +-2/3, +-1, +-2, +-3, +-6, +-9, +-18#

None of these work, so #f(x)# has no rational zeros.

In common with most quintics and higher order polynomials, the zeros are not expressible in terms of #n#th roots or elementary functions, including trigonometric ones.

About the best you can do is use a numerical method like Durand-Kerner to find approximations:

#x_1 ~~ -1.35047#

#x_(2,3) ~~ -0.506174+-1.35526i#

#x_(4,5) ~~ 1.18141+-0.852685i#

See https://socratic.org/s/awNxzXZ9 for a description of the method and another example quintic approximated using this method.