How do you use the remainder theorem to find P(c) `p(x)=x^4+3x^3-4x^2+6x-31`; 2?

`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`