According to half interval method, let #epsilon<0.005# the error which is permitted (here we have approximation to #2# decimal places, so choosing it accordingly).
Now, find two numbers #alpha,beta#, for which #f(x)# has different sign, then define #gamma=(alpha+beta)/2#. If #beta-gamma < epsilon#, then accept #gamma# as root, otherwise if #f(alpha)# and #f(gamma)# are of different sign i.e. #f(alpha)f(gamma) < 0#, then replace #alpha# or #beta# with #gamma#, so that they are of different sign.
Here for #f(x)=x^3-8x+1#, #f(-3)=-2# and #f(-2.8)=1.448#.
So let us choose #-2.9# for which #f(-2.9)=-0.189# and our new #alpha,beta# are #-2.8# and #-2.9#. Now selecting at half interval #-2.85#, we have #f(-2.85)=0.650875#.
Our new #alpha,beta# are #-2.85# and #-2.9# and selecting at half interval #-2.8875#, we have #f(-2.875)=0.236328# and now we choose between #-2.875# and #-2.9# i.e. #-2.8875# and #f(-2.8875)=0.025018#.
Similarly we now choose between #-2.8875# and #-2.9# i.e. #-2.89375# and as #f(-2.89375)=-0.08165# and now we choose between #-2.89375# and #-2.8875# i.e. #-2.890625# and #f(-2.896875)=-0.028233#
We now choose between #-2.896875# and #-2..8875# i.e. #-2.8921875# and as #f(-2.8921875)=-0.054921#, we now choose between #-2.8921875# and #-2.8875# i.e. #-2.88984375# and #f(-2.88984375)=-0.014904#
Hence, root of #x^3-8x+1# approximate to two places of decimal is #-2.89#
