# What are all the possible rational zeros for #f(x)=x^3-13x^2+23x-11# and how do you find all zeros?

##### 1 Answer

The "possible" rational zeros are:

The actual zeros are:

#### Explanation:

#f(x) = x^3-13x^2+23x-11#

By the rational roots theorem, any rational zeros of

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

#+-1, +-11#

Note that the sum of the coefficients of

#1-13+23-11 = 0#

Hence

#x^3-13x^2+23x-11 = (x-1)(x^2-12x+11)#

The sum of the coefficients of the remaining quadratic is zero too:

#1-12+11 = 0#

So

#(x^2-12x+11) = (x-1)(x-11)#

So the zeros of