Given: #(3x^4 - 3x^3 + x^2 + x - 15)/(x-3)^2#
Using long division, the denominator becomes: #x^2 - 6x + 9#
#x^2 - 6x + 9 | bar(3x^4 - 3x^3 + x^2 + x - 15)#
We want to eliminate #3x^4#. What times #x^2 = 3x^4#? #" "3x^2#
Multiply #3x^2# times each of the terms in the divisor and then subtract:
#" "3x^2#
#x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + x - 15)#
#" "ul(3x^4 - 18x^3 + 27x^2)#
#" "15x^3 - 26x^2#
Bring down the next term from the dividend:
#" "3x^2#
#x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + x - 15)#
#" "ul(3x^4 - 18x^3 + 27x^2)#
#" "15x^3 - 26x^2 + x#
We want to eliminate #15x^3#. What times #x^2 = 15x^3#? #" "15x#
Multiply #15x# times each of the terms in the divisor and then subtract and then bring down the next dividend term:
#" "3x^2 + 15x#
#x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + " "x - 15)#
#" "ul(3x^4 - 18x^3 + 27x^2)#
#" "15x^3 - 26x^2 + " "x#
#" "ul(15x^3 -90x^2 + 135x)#
#" "64x^2 - 134x - 15#
We want to eliminate #64x^2#. What times #x^2 = 64x^2#? #" "64#
Multiply #64# times each of the terms in the divisor and then subtract
#" "3x^2 + 15x + 64#
#x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + " "x - 15)#
#" "ul(3x^4 - 18x^3 + 27x^2)#
#" "15x^3 - 26x^2 + " "x#
#" "ul(15x^3 -90x^2 + 135x)#
#" "64x^2 - 134x - 15#
#" "ul(64x^2 - 384x + 576#
#" "250x - 591#
#(3x^4 - 3x^3 + x^2 + x - 15)/(x-3)^2 = 3x^2+15x+64 + (250x-591)/(x-3)^2#
