# What are all the possible rational zeros for #f(x)=3x^3-11x^2+5x+3# and how do you find all zeros?

##### 1 Answer

#### Explanation:

#f(x) = 3x^3-11x^2+5x+3#

By the rational roots theorem, any *rational* zero of

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

#+-1/3, +-1, +-3#

Note first that there's a shortcut in this particular example, in that the sum of the coefficients is

Hence we can deduce that

#3x^3-11x^2+5x+3 = (x-1)(3x^2-8x-3)#

We could just try the remaining rational possibilities that we listed, but we can factor the remaining quadratic (

Find a pair of factors of

The pair

Use this pair to split the middle term and factor by grouping:

#3x^2-8x-3 = 3x^2-9x+x-3#

#color(white)(3x^2-8x-3) = (3x^2-9x)+(x-3)#

#color(white)(3x^2-8x-3) = 3x(x-3)+1(x-3)#

#color(white)(3x^2-8x-3) = (3x+1)(x-3)#

Hence the other two zeros of