How do you find all the zeros of x5−3x4+5x3+9x2+x−2?

1 Answer
George C.
Jun 10, 2016

Use a numerical method to find approximations:

x1≈0.390503

x2,3≈2.01006±2.24677i

x4,5≈−0.705308±0.257059i

Explanation:

f(x)=x5−3x4+5x3+9x2+x−2

By the rational root theorem, any rational zeros of f(x) are expressible in the form pq 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 nth 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:

x1≈0.390503

x2,3≈2.01006±2.24677i

x4,5≈−0.705308±0.257059i

I used the following C++ program:

enter image source here

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