Question

Find the remainder when x to the power of 4 +xcube - 2x square + x+1 is divided by x+1

Answer 1

Correct Question -

Find the remainder when -

$\sf{ f(x) = {x}^4 + {x}^3 - 2{x}^2 + x + 1 }$ is divided by x + 1 .

Solution -

Here, in the above Question , we have been given the following information -

$\sf{ f(x) = {x}^4 + {x}^3 - 2{x}^2 + x + 1 }$ is divided by x + 1 .

Let me first define the reminder Theorem ..

Remainder Theorem -

According to the remainder theorem , a polynomial, f(x), is divided by a linear polynomial , x + a, then the required remainder will be f(-a).

Here ,

$\sf{ f(x) = {x}^4 + {x}^3 - 2{x}^2 + x + 1 }$

$\sf{ Linear \ Polynomial \ - \ x + 1 }$

So,

Remainder = f ( -1 )

$\sf{ f(x) = {x}^4 + {x}^3 - 2{x}^2 + x + 1 } \\ \sf{ f( -1) = {-1}^4 + {-1}^3 - 2{ -1 }^2 + { -1 } + 1 } \\ \\ => \sf{ 1 - 1 - 2 - 1 + 1 } \\ \\ \sf{ -2}$

Hence the required remainder is -2.

_____________