According to remainder theorem when we divide a polynomial P(x)P(x) (here P(x)=6x^3-x^2+4x+3P(x)=6x3−x2+4x+3), by a binomial of degree one say (x-a)(x−a), (here (x-3)(x−3) as we desire P(a)P(a) wih a=3a=3)
the remainder is P(a)P(a)
Normally to find P(a)P(a), one substitute xx with aa, but as we have to use Remainder theorem, we should divide P(x)P(x) by (x-a)(x−a) and then the remainder would be P(a)P(a). For this let us use synthetic division to divide 6x^3-x^2+4x+36x3−x2+4x+3 by (x-3)(x−3).
3|color(white)(X)6" "color(white)(X)-1color(white)(XXX)4" "" "33∣X6 X−1XXX4 3
color(white)(1)|" "color(white)(XXxx)18color(white)(XXX)51color(white)(XX)1651∣ XX×18XXX51XX165
" "stackrel("—————————————)
color(white)(x)|color(white)(X)color(blue)6color(white)(Xxxx)color(blue)17color(white)(XXX)color(blue)55color(white)(XX)color(red)168
i.e. while quotient is 6x^2+17x+55, remainder is 168.
