# How do you factor #z^3-z^2-13z+4#?

##### 1 Answer

#### Answer:

#### Explanation:

By the rational root theorem, any rational zeros must be expressible in the form

That means that the only possible rational zeros are:

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

Let

We find

#z^3-z^2-13z+4 = (z-4)(z^2+3z-1)#

We can factor the remaining quadratic factor by completing the square:

#z^2+3z-1#

#= (z+3/2)^2-9/4+1#

#= (z+3/2)^2-5/4#

#= (z+3/2)^2-(sqrt(5)/2)^2#

#= (z + 3/2 - sqrt(5)/2)(z + 3/2 + sqrt(5)/2)#

Putting it all together:

#z^3-z^2-13z+4 = (z-4)(z + 3/2 - sqrt(5)/2)(z + 3/2 + sqrt(5)/2)#