#p(x) = 3x^3-2x^2+6x-4#
The Remainder Theorem states that when we divide a polynomial #f(x)# by #x-c# the remainder #R# equals #f(c)#.
We use synthetic substitution to divide #f(x)# by #x-c#, where #c = -4#.
Step 1. Write only the coefficients of #x# in the dividend inside an upside-down division symbol.
#|3" "-2" " "6" " " "color(white)(1)-4#
#|color(white)(1)#
#stackrel("—————————————)#
Step 2. Put the divisor at the left.
#color(red)(-4)|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|" "#
#" "" "stackrel("—————————————)#
Step 3. Drop the first coefficient of the dividend below the division symbol.
#-4|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|color(white)(1)#
#" "" "stackrel("—————————————)#
#" "" "color(red)(3)#
Step 4. Multiply the drop-down by the divisor, and put the result in the next column.
#-4|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|" " "color(white)(1)color(red)(-12)#
#" "" "stackrel("—————————————)#
#" "" "3#
Step 5. Add down the column.
#-4|3" "-2" " "6" " " "color(white)(1)-4#
#" "color(white)(1)|" " "-12#
#" "" "stackrel("—————————————)#
#" "" "3" "color(white)(1)color(red)(-14)#
Step 6. Repeat Steps 4 and 5 until you can go no farther.
#-4|3" "-2" "color(white)(1)6" " "color(white)(1)-4#
#" "color(white)(1)|" " " " "-12" " 56" "color(white)(1)-248#
#" "" "stackrel("—————————————)#
#" "" "3" "-14color(white)(1)62" "color(red)(-252)#
The remainder is #-252#, so #p(-4) = -252#.
Check:
#3x^3-2x^2+6x-4 = 3(-4)^3-2(-4)^2+6(-4)-4 = 3(-64)-2(16) -24-4= -192-32-28=-252#
