# How do you use the rational roots theorem to find all possible zeros of #P(x) = x^5 -2x^4 +4x -8#?

##### 1 Answer

The only Real zero is

#### Explanation:

By the rational roots theorem, any rational zeros of

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

#+-1# ,#+-2# ,#+-4# ,#+-8#

Trying each in turn, we find:

#P(2) = 32-32+8-8 = 0#

So

#x^5-2x^4+4x-8 = (x-2)(x^4+4)#

Note that

It does have Complex zeros

**Bonus**

Note that although

In fact, we find:

#x^4+4 = (x^2-2x+2)(x^2+2x+2)#

This is an instance of a nice identity for factoring quartics:

#(a^2-kab+b^2)(a^2+kab+b^2) = a^4+(2-k^2)a^2b^2+b^4#

Note in particular that if we put

#(a^2-sqrt(2)ab+b^2)(a^2+sqrt(2)ab+b^2) = a^4+b^4#

In our case we used this with