How do you find the quotient of #(x^3+3x^2-3x-2)div(x-1)# using long division?

1 Answer
Apr 23, 2018

# x^3 + 3x^2 - 3x - 2 = ( x -1)(x^2 + 4x + 1) - 1 #

Explanation:

# text{ -------------------------} #
# x -1 quad text{)} quad x^3 + 3x^2 - 3x - 2 #

That's a pain to format. Anyway, the first "digit", first term in the quotient, is #x^2#. We compute the digit times #x-1#, and take that away from #x^3 + 3x^2 - 3x -2 #:

#text{ } x^2#
# text{ -------------------------} #
# x -1 quad text{)} quad x^3 + 3x^2 - 3x - 2 #
# text{ } x^3 -x^2 #
# text{ ----------------} #
# text{ } 4 x^2 - 3x - 2#

OK, back to the quotient. The next term is #4x# because that times #x# gives #4 x^2#. After that the term is #1#.

#text{ } x^2 + 4 x + 1#
# text{ ------------------------- #
# x -1 quad text{)} quad x^3 + 3x^2 - 3x - 2 #
# text{ } x^3 -x^2 #
# text{ ----------------} #
# text{ } 4 x^2 - 3x - 2#
# text{ } 4 x^2 - 4x#
# text{ ----------------} #
# text{ } x - 2 #
# text{ } x - 1 #
# text{ --------} #
# text{ } -1#

We have a non-zero remainder! That says

# x^3 + 3x^2 - 3x - 2 = ( x -1)(x^2 + 4x + 1) - 1 #