# How do you factor given that #f(4)=0# and #f(x)=x^3-14x^2+47x-18#?

##### 1 Answer

#### Explanation:

The difference of squares identity can be written:

#a^2-b^2 = (a-b)(a+b)#

We use this later with

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

So the only possible *rational* zeros are:

#+-1, +-2, +-3, +-6, +-9, +-18#

In particular

Trying each of these in turn, we (eventually) find:

#f(9) = 729-1134+423-18 = 0#

So

#x^3-14x^2+47x-18#

#= (x-9)(x^2-5x+2)#

#= (x-9)((x-5/2)^2-25/4+2)#

#= (x-9)((x-5/2)^2-(sqrt(17)/2)^2)#

#= (x-9)(x-5/2-sqrt(17)/2)(x-5/2+sqrt(17)/2)#