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
