From the given (-3x^3+12x^2-14x-6)/(x+1)
We have our
dividend -3x^3+12x^2-14x-6
divisor x+1
Let us begin the
color(red)("Long Division Method")
" " " " " "underline(-3x^2+15x-29)larrthe quotient
x+1|~-3x^3+12x^2-14x-6
" " " " " "underline(-3x^3-3x^2" " " " " " " " " " ")
" " " " " " " " " 0"" +15x^2-14x-6
" " " " " " " " " " " "underline(15x^2+15x" " " " ")
" " " " " " " " " " " " " "0"" -29x-6
" " " " " " " " " " " " " " " " underline(-29x-29)
" " " " " " " " " " " " " " " " " " " 0""+23 larrthe remainder
We write our final answer
(-3x^3+12x^2-14x-6)/(x+1)=-3x^2+15x-29+23/(x+1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(red)("Another method is by Synthetic Division")
Our trial divisor comes from the divisor x+1. Simply equate this to zero
x+1=0 and x=-1
Use the numerical coefficients of the dividend
" "x^3" " " "" "x^2" " " " " "x^1" " " " " "x^0
-3" " " "+12" " " "-14" " " "-6" " "|~-1larrtrial divisor
underline(" " " " " " " "+3" " " "-15" " " "+29" " )
-3" " " "+15" " " "-29" " " "+23larr the remainder
The last set of numbers -3, +15, -29 are the numerical coefficients of x^2, x^1, x^0 repectively so that our quotient is
-3x^2+15x-29
since the remainder is +23, we write the last term (+23)/(x+1)
Our final answer is
(-3x^3+12x^2-14x-6)/(x+1)=-3x^2+15x-29+23/(x+1)
God bless....I hope the explanation is useful.
