What are all the possible rational zeros for f(x)=x^5-6x^4+12x^2-8x+36 and how do you find all zeros?

1 Answer
Aug 27, 2016

Find there are no rational zeros. Use a numerical method to find approximations.

Explanation:

f(x) = x^5-6x^4+12x^2-8x+36

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 36 and q a divisor of the coefficient 1 of the leading term.

That means that the only possible rational zeros are:

+-1, +-2, +-3, +-4, +-6, +-9, +-12, +-18, +-36

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

About the best we can do is use the Durand-Kerner or similar numerical method to find approximations:

x_1 ~~ 5.63048

x_2 ~~ 2.05347

x_3 ~~ -1.84949

x_(4,5) ~~ 0.0827684+-1.29486i

See https://socratic.org/s/axpvv6Hk for some more description of the method.

For the current example, the above approximations were found using this C++ program...

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