# \ #
# "When" \ f(x) \ "is divided by" \ x - 2, \ "the remander is given to be" \ \ 1. #
# "So we may write:" #
# \qquad \quad f(x) \ = \ q(x) ( x - 2 ) \ + \ 1; \qquad "for some polynomial" \ \ q(x). #
# "Squaring both side of the previous equation, we get:" #
# \qquad \qquad \qquad \qquad \qquad \quad \ \ f(x)^2 \ = \ [ q(x) ( x - 2 ) \ + \ 1 ]^2 #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad= \ [ q(x) ( x - 2 ) ]^2 \ + \ 2 \cdot ( x - 2 ) + [ 1 ]^2 #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ q(x)^2 ( x - 2 )^2 \ + \ 2 \cdot ( x - 2 ) + 1 #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ [ q(x)^2 ( x - 2 ) + 2 ] ( x - 2 ) + 1 #
# \qquad \qquad = \ Q(x) ( x - 2 ) + 1; \qquad "where" \quad Q(x) = \ q(x)^2 ( x - 2 ) + 2. #
# \qquad \ :. \qquad f(x)^2 \ = \ Q(x) ( x - 2 ) + 1; \qquad "for polynomial" \quad Q(x). #
# "From this last equation, we read off that the remainder there" #
# "is :" \ 1. #
# \ #
# "So we conclude:" #
# \qquad \quad "the remainder when" \ f(x)^2 \ "is divided by" \ \ x - 2 \ \ "is:" \quad \quad 1. #
