Given: #(x^4-2x^3+2x^2-3x+3)/(x^2-3)#
Write the divisor with a 0 coefficient for missing term
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))color(white)((x^4-2x^3+2x^2-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
To determine the first term in the quotient divide the first term in the dividend by the first term in the divisor #x^4/x^2=x^2# write #x^2# in the quotient:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2color(white)(2x^3+2x^2-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
Multiply the term in the quotient by the divisor:
#x^2(x^2+0x-3) =x^4+0x^3-3x^2#
Make it negative:
#-x^4-0x^3+3x^2#
Write it below the dividend:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2color(white)(2x^3+2x^2-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
Perform the addition and bring down the x term:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2color(white)(2x^3+2x^2-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
#color(white)(..........................)-2x^3+5x^2-3x#
To find the next term in the quotient, divide the first term in the sum by the first term of the divisor:
#(-2x^3)/x^2=-2x#
Write it as the next term of the quotient:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2-2xcolor(white)(2x^2-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
#color(white)(..........................)-2x^3+5x^2-3x#
Multiply the term in the quotient by the divisor:
#-2x(x^2+0x-3) =-2x^3+0x^2+6x#
Make it negative:
#2x^3+0x^2-6x#
Write it below the sum:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2-2xcolor(white)(2x^2-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
#color(white)(..........................)-2x^3+5x^2-3x#
#color(white)(...............................)ul(2x^3+0x^2-6x)#
Perform the additions and bring down the constant term:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2-2xcolor(white)(2x^2-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
#color(white)(..........................)-2x^3+5x^2-3x#
#color(white)(...............................)ul(2x^3+0x^2-6x)#
#color(white)(..........................................)5x^2-9x+3#
To find the next term in the quotient, divide the first term in the sum by the first term of the divisor:
#(5x^2)/x^2=5#
Write it as the next term of the quotient:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2-2x+5color(white)(-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
#color(white)(..........................)-2x^3+5x^2-3x#
#color(white)(...............................)ul(2x^3+0x^2-6x)#
#color(white)(..........................................)5x^2-9x+3#
Multiply the term in the quotient by the divisor:
#5(x^2+0x-3)=5x^2+0x-15#
Make it negative:
#-5x^2+0x+15#
Write it below the sum:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2-2x+5color(white)(-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
#color(white)(..........................)-2x^3+5x^2-3x#
#color(white)(...............................)ul(2x^3+0x^2-6x)#
#color(white)(..........................................)5x^2-9x+3#
#color(white)(.......................................)ul(-5x^2+0x+15)#
Perform the addition:
#color(white)( (x^2+0x-3)/color(black)(x^2+0x-3))(x^2-2x+5color(white)(-3x+3))/(| color(white)(x)x^4-2x^3+2x^2-3x+3)#
#color(white)(....................)ul(-x^4-0x^3+3x^2)#
#color(white)(..........................)-2x^3+5x^2-3x#
#color(white)(...............................)ul(2x^3+0x^2-6x)#
#color(white)(..........................................)5x^2-9x+3#
#color(white)(.......................................)ul(-5x^2+0x+15)#
#color(white)(...............................................)-9x+18#
Because #-9x+18# is of order 1 and the divisor is of order 2, we stop and declare #-9x+18# to be the remainder.
