Question

list all rational zeros for f(x)=2x^3+5x^2-x-1. Now I know the possible rational zeros are 1,-1,(1/2),(-1/2). I cant find the rational zero for x using synthetic division.

Answer 1

I would first try -1 and +1 because they are easy to figure out:
for x=-1
-2 + 5 +1 -1 = +3

for x= 1:
2+5-1-1= 5

for x=0 we get -1
because the function is -1 at 0 and goes positive (+5) at x=1, it crosses the x-axis between 0 and 1. So try x= 1/2

2*1/8 + 5*1/4 -1/2 - 1
this simplifies to
1/4 + 5/4 - 2/4 -4/4 or
6/4 - 6/4 =0

so x= 1/2 is the rational root.

if you use synthetic division, you will find
2(x^2 + 3x + 1)
which has no rational roots. so x=1/2 is the only rational root.