We can perform the Long Division and get the Quotient and the
Remainder. But, here is another way to solve the Problem.
Suppose that, when #P(x)=x^3-2x^2-3x+2# is divided by
#(x-1)#, the Quotient Poly. is #Q(x)# and the remainder #R.# Note
that, since the divisor #9x-1)# is a Linear Poly., the Remainder has
to be a constant.
The well-known relation btwn. #P(x), Q(x), (x-1) and, R# is
given by, #P(x)=(x-1)Q(x)+R,# i.e.,
# x^3-2x^2-3x+2=(x-1)Q(x)+R........................(star)#
Sub.ing, #x=1" in "(star), 1-2-3+2=(1-1)Q(x)+R :. R=-2.#
Then, sub.ing #R=-2" in "(star), x^3-2x^2-3x+2=(x-1)Q(x)-2,#
#or, x^3-2x^2-3x+4=(x-1)Q(x)#
# rArr Q(x)=(x^3-2x^2-3x+4)/(x-1)#
#=(x^3-x^2-x^2+x-4x+4)/(x-1)#
#={x^2(x-1)-x(x-1)-4(x-1)}/(x-1)#
#={cancel((x-1))(x^2-x-4)}/cancel((x-1))#
#:. Q(x)=x^2-x-4#.
Enjoy Maths.!
