# How do you find the zeros of #f(x) = 4x^5 + x^4 + x^3 + x^2 - 2x - 2#?

##### 1 Answer

Numerically...

#### Explanation:

Given:

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

By the rational roots theorem, any rational zeros of

That means that the only possible rational zeros are:

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

None of these is a zero and hence we can tell that

Note that the signs of the coefficients are in the pattern

The signs of the coefficients of

In fact it has two complex conjugate pairs of non-real complex zeros.

Typically for a quintic polynomial none of these zeros is expressible in terms of

Using a Durand-Kerner method, I found approximate zeros:

#x_1 ~~ 0.854463#

#x_(2,3) ~~ 0.144877+-1.00403i#

#x_(4,5) ~~ -0.697109+-0.287521i#

Here's the C++ program I used:

See https://socratic.org/s/aSVeQs9C for another example and a fuller description of this method.