can anyone help me??

Question

f(x)=5x3-29x²+55x-28

find all the rational zeros of each function.

Answer 1

The rational zeros theorem is a method to determine the possible roots of a function.  The rational zeros are p/q, where p are all of the factors of the constant term, and q are all of the factors of the leading coefficient. 

For this function, p = +/- 28, +/- 14, +/- 7, +/- 4, +/- 2, +/-1.

And q = +/- 5, +/- 1.  

The rational zeros are therefore (all +/-) 28/5, 14/5, 7/5, 4/5, 2/5, 1/5, 28, 14, 7, 4, 2, 1.  

Determining that the actual root is in fact 4/5 is tedious and requires applying synthetic division to each of these rational zeros.  

Answer 2

You can factor this as f(x) = (5x-4)(x2-5x+7)

So 5x-4 =0 gives x=(4/5). This is a real rational root.

x2-5x+7=0 use the quadratic formula to find that x= (5+i*sqrt(3))/2 and x =(5-i*sqrt(3))/2. These are both complex irrational roots.

The  only rational zero is x = (4/5).