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
