# How do you find all the real and complex roots and use Descartes Rule of Signs to analyze the zeros of # P(x) = 2x^5 + 7x^3 + 6x^2 - 2 #?

##### 1 Answer

See explanation...

#### Explanation:

#P(x)=2x^5+7x^3+6x^2-2#

The pattern of signs of the coefficients is

#P(-x) = -2x^5-7x^3+6x^2-2#

The pattern of signs of the coefficients of

By the fundamental theorem of algebra,

By the rational root theorem, any *rational* zeros of

That means that the only possible *rational* zeros are:

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

None of these work, so *rational* zeros.

In common with most quintics and higher order polynomials, the zeros of

About the best you can do is use a numerical method such as Durand-Kerner to find approximations for the zeros:

#x_1 ~~ 0.460754#

#x_(2,3) ~~ -0.617273+-0.377799i#

#x_(4,5) ~~ 0.386896+-1.99853i#