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
