# How do you list all possible roots and find all factors and zeroes of #4x^3-9x^2+6x-1#?

##### 1 Answer

with zeros

#### Explanation:

#f(x) = 4x^3-9x^2+6x-1#

I notice that the question asks for possible roots, so you are probably expected to make use of the rational root theorem first:

Since this cubic is given in standard form (with descending powers of

Any *rational* zeros of

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

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

If we evaluate

#f(1/4) = 4/64-9/16+6/4-1 = (1-9+24-16)/16 = 0#

So

#4x^3-9x^2+6x-1#

#= (4x-1)(x^2-2x+1)#

#= (4x-1)(x-1)^2#

Hence we have zeros:

#x=1/4#

#x=1# with multiplicity#2#

**Footnote**

If the question did not mention "possible" roots, then I would have found the solution by looking at the sum of the coefficients first:

Note that

#4x^3-9x^2+6x-1 = (x-1)(4x^2-5x+1)#

Then note that

#4x^2-5x+1 = (x-1)(4x-1)#

Putting it together:

#4x^3-9x^2+6x-1 = (x-1)(x-1)(4x-1)#