P(x) = x^4+3x^3-4x^2+6x-31.
The Remainder Theorem tells us that if we divide P(x), by x-c, then the remainder is P(c).
So we'll divide P(x) by x-2. You could use long division, but synthetic division requires less writing and we can use it for any linear divisor with leading coefficient 1, so:
"2" || "1 " " " "3 " " " "-4 " " " "6" " " "-31
+ "" " " " " "" "2" " " " 10" " " "12" " " " " "36"
"" " "---------
"" " " " 1" " " " 5" " "" " "6" " " "18" " " "||" " 5"
The remainder is 5, so P(2) = 5
If you use long division, it will look something like:
" " " " " " " x^3 +5x^2+6x +18
" " " "-----------
x-2 ) x^4 +3x^3 -4x^2 +6x -31
" " " " x^4 -2x^3
" " " "-----------
" " " " " "" " 5x^3" " -4x^2 +6x -31
" " " " " "" " 5x^3" " -10x^2
" " " "-----------
" " " " " " " " " "" " " " " " 6x^2 +6x -31
" " " " " " " " " "" " " " " " 6x^2 -12x
" " " "-----------
" " " " " " " " " "" " " " " " " " " " 18x -31
" " " " " " " " " "" " " " " " " " " " 18x -36
" " " "-----------
" " " " " " " " " "" " " " " " " " " " " " " " " " 5
Again, the remainder is 5, so P(2) =5
