Question

What is the remainder when `x^4-3x^2+7x+3` is divided by `x-2`?

Answer 1

`"remainder "=21`

Explanation:

`"one way of dividing is to use the divisor as a factor in "`
`"the numerator"`

`"consider the numerator"`

`color(red)(x^3)(x-2)color(magenta)(+2x^3)-3x^2+7x+3`

`=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(magenta)(+4x^2)-3x^2+7x+3`

`=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(red)(+x)(x-2)color(magenta)(+2x)+7x+3`

`=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(red)(+x)(x-2)color(red)(+9)(x-2)color(magenta)(+18)+3`

`=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(red)(+x)(x-2)color(red)(+9)(x-2)+21`

`"quotient "=color(red)(x^3+2x^2+x+9)," remainder "=21`

Answer 2

`21`

Explanation:

We can use the Remainder theorem , which states;

`"For the Polynomial "P(x) " its remainder on division by "`

`(x-a) " is "P(a)`

proof

`(x-a)" a factor of "P(x)`

`=>P(x)=(x-a)Q(x)+R---(1)`

where

`Q(x)" is the quotient polynomial"`

`R=" the remainder"`

taking `(1)`

`P(a)=cancel((a-a)Q(a))+R`

`:.P(a)=R`

so we have

`P(x)=x^4-3x^2+7x+3`

it is to be divided by

`(x-2)`

`:. " we require "P(2)`

`P(2)=2^4-3xx2^2+7xx2+3`

`P(2)=16-12+14+3`

`P(2)=21`