The Remainder Theorem states that when we divide a polynomial #f(x)# by #x-c# the remainder #R# equals #f(c)#.
We use synthetic division to divide #f(x)# by #x-c#, where #c = -3#.
Step 1. Write only the coefficients of #x# in the dividend inside an upside-down division symbol.
#|1" " -4" " " " "2"" "-4" "+6#
#|color(white)(1)#
#stackrel("————————————————————)#
Step 2. Put the divisor at the left.
#color(red)(-3)|1" " -4" " " " "2"" "-4" "+6#
#" " "|#
#" " " "stackrel("————————————————————)#
Step 3. Drop the first coefficient of the dividend below the division symbol.
#-3|1" " -4" " " " "2"" "-4" "+6#
#" " " |#
#" "stackrel("————————————————————)#
#" " " "color(red)(1)#
Step 4. Multiply the drop-down by the divisor, and put the result in the next column.
#-3|1" " -4" " " " "2"" "-4" "6#
#" " " |" " " " "color(red)(-3)#
#" " " "stackrel("————————————————————")#
#" " " "1#
Step 5. Add down the column.
#-3|1" " -4" " " " "2"" "-4" "6#
#" " " |" " " " "-3#
#" "stackrel("————————————————————)#
#" " " "1" " " "color(red)(-7)#
Step 6. Repeat Steps 4 and 5 until you can go no farther
#-3|1" " -4" " " " "2"" "-4" " " " " "6#
#" " " |" " " " "-3" "+21 -69" " " "219#
#" "stackrel("————————————————————)#
#" " " "1" " -7" " " " 23" "-73 " " "color(red)(225)#
The remainder is #225#, so #f(-3) = 225#.
Check:
#x^4-4x^3 +2x^2 –4x +6 = (-3)^4-4(-3)^3+2(-3)^2-4(-3)+6 = 81+108 +18+12+6 = 225#
