# How do you find all the real and complex roots of #2x^5-4x^4-4x^2+5=0#?

##### 1 Answer

Use a numerical method to find approximations:

#x ~~ 2.2904#

#x ~~ 0.925274#

#x ~~ -0.80773#

#x ~~ -0.203974+-1.19116i#

#### Explanation:

**Fundamental theorem of algebra**

The FTOA tells us that any non-zero polynomial of degree

In our example

**Rational root theorem**

SInce *rational* zeros must be expressible in the for

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

#+-1/2, +-1, +-5/2, +-5#

None of these satisfy *rational* zeros.

**Descartes rule of signs**

The signs of the coefficients of

The signs of the coefficients of

That leaves

**Quintic**

In common with most quintics (and higher degree polynomials), the zeros of this one cannot be expressed using elementary functions like

#x ~~ 2.2904#

#x ~~ 0.925274#

#x ~~ -0.80773#

#x ~~ -0.203974+-1.19116i#

See https://socratic.org/s/avVFw8eC for more details of the method and another example quintic.