#p(x) = x^3-4x^2+4x-5#
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.
#|1" "-4" " "4" " " "-5#
#|color(white)(1)#
#stackrel("—————————————)#
Step 2. Put the divisor at the left.
#color(red)(4)|1" "-4" " "4" " " "-5#
#color(white)(1)|color(white)(1)#
#" "stackrel("—————————————)#
Step 3. Drop the first coefficient of the dividend below the division symbol.
#4|1" "-4" " "4" " " "-5#
#color(white)(1)|" "" "color(white)(1)#
#" "stackrel("—————————————)#
#" "color(white)(1)color(red)(1)#
Step 4. Multiply the drop-down by the divisor, and put the result in the next column.
#4|1" "-4" " "4" " " "-5#
#color(white)(1)|" "" "color(white)(1)color(red)(4)#
#" "stackrel("—————————————)#
#" "color(white)(1)1#
Step 5. Add down the column.
#4|1" "-4" " "4" " " "-5#
#color(white)(1)|" "" "color(white)(1)4#
#" "stackrel("—————————————)#
#" "color(white)(1)1" "" "color(red)(0)#
Step 6. Repeat Steps 4 and 5 until you can go no farther
#4|1" "-4" " "4" " " "-5#
#color(white)(1)|" "" "color(white)(1)4" "0" "" "16#
#" "stackrel("—————————————)#
#" "color(white)(1)1" "" "0" "4" "" "color(red)(11)#
The remainder is #11#, so #p(4) = 11#.
Check:
#p(x) = x^3-4x^2+4x-5#
#p(4) = 4^3-4(4)^2+4(4)-5 = 64-4(16)+16-5= 64-64-11 = 11#
