How do you use the rational roots theorem to find all possible zeros of #f(x)=x^5-3x^2-4#?

1 Answer
Aug 13, 2016

Answer:

Find #f(x)# has no rational zeros.

We can find numerical approximations:

#x_1 ~~ 1.6477#

#x_(2,3) ~~ 0.151952+-1.03723i#

#x_(4,5) ~~ -0.975801+-1.12111i#

Explanation:

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#f(x) = x^5-3x^2-4#

By the rational roots 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 #-4# and #q# a divisor of the coefficient #1# of the leading term.

That means that the only possible rational zeros are:

#+-1, +-2, +-4#

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

In common with most quitics and polynomials of higher degree, the zeros of this one cannot be expressed in terms of #n#th roots and/or elementary functions, including trigonometric or exponential ones.

It is possible to find numerical approximations for the zeros using a method like Durand-Kerner.

For example, we can use this C++ program...

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to find numerical approximations for the zeros:

#x_1 ~~ 1.6477#

#x_(2,3) ~~ 0.151952+-1.03723i#

#x_(4,5) ~~ -0.975801+-1.12111i#

For a little more explanation of the method see https://socratic.org/s/ax2iiWhR