#color(red)(f(-2) = 70)#

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 = -2#.

**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)(-2)|1" " -4" " "2" " " " "-4" " " " "6#

#" " " |#

#" "stackrel("————————————————————)#

**Step 3.** Drop the first coefficient of the dividend below the division symbol.

#-2|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.

#-2|1" " -4" " " " "2"" "-4" "6#

#" " " |" " " " "color(red)(-2)#

#" " " "stackrel("————————————————————")#

#" " " "1#

**Step 5.** Add down the column.

#-3|1" " -4" " " " "2"" "-4" "6#

#" " " |" " " " "-2#

#" " " "stackrel("————————————————————)#

#" " " "1" " " "color(red)(-6)#

**Step 6.** Repeat Steps 4 and 5 until you can go no farther

#-2|1" " -4" " "2" " " " "-4" " " " "6#

#" " " |" " " " "-2" "12" "-28" " " " "64#

#" " " "stackrel("————————————————————)#

#" " " "1" "-6" " 14" "-32 " " " "color(red)(70)#

The remainder is #70#, so #f(-2) = 70#.

**Check:**

#x^4-4x^3 +2x^2 –4x +6 = (-2)^4-4(-2)^3+2(-2)^2-4(-2)+6 = 16+32 +8+8+6 = 70#