According to remainder theorem if a polynomial #f(x)# is divided by #(x-p)#, the remainder is #f(p)#.
Hence, using remainder theorem to evaluate #f(a)=a^4+3a^3-17a^2+2a-7# at #a=3#,
we should divide #a^4+3a^3-17a^2+2a-7# by #(a-3)#
this can be done using synthetic division
#3|color(white)(X)1" "color(white)(X)3color(white)(XX)-17" "" "2color(white)(XX)-7#
#color(white)(x)|" "color(white)(Xxx)3color(white)(XXX)18color(white)(Xxxx)3color(white)(Xxxx)15#
#" "stackrel("—————————————----)#
#color(white)(x)|color(white)(X)color(blue)1color(white)(X11)color(red)6color(white)(XXXX)color(red)1color(white)(XXX)color(red)5color(white)(Xxxx)8#
Hence, Quotient is #a^3+6a^2+a+5# and remainder is #8#.
Hence #f(a)=a^4+3a^3-17a^2+2a-7# at #a=3# is #8#
