How do you find all the zeros of #x^5 - 3x^4 + 5x^3 + 9x^2 + x - 2#?

1 Answer
Jun 10, 2016

Answer:

Use a numerical method to find approximations:

#x_1 ~~ 0.390503#

#x_(2,3) ~~ 2.01006+-2.24677i#

#x_(4,5) ~~ -0.705308+-0.257059i#

Explanation:

#f(x) = x^5-3x^4+5x^3+9x^2+x-2#

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 #-2# and #q# a divisor of the coefficient #1# of the leading term.

So the only possible rational zeros are:

#+-1#, #+-2#

Neither of these is a zero, so #f(x)# has no rational zeros.

In common with quintics in general, this #f(x)# has no algebraic solution in terms of #n#th roots.

We can find rational approximations using a numeric method such as Durand-Kerner. For another example of such a quintic solution, see: https://socratic.org/s/avdSNDdg

In the current example we find approximations:

#x_1 ~~ 0.390503#

#x_(2,3) ~~ 2.01006+-2.24677i#

#x_(4,5) ~~ -0.705308+-0.257059i#

I used the following C++ program:

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