What must be added to the polynomial $f(x)\ =\ x^4\ +\ 2x^3\ -\ 2x^2\ +\ x\ -\ 1$ so that the resulting polynomial is exactly divisible by $x^2\ +\ 2x\ -\ 3$?


Given: 

Given polynomial is $f(x)\ =\ 3x^4\ -\ 9x^3\ +\ x^2\ +\ 15x\ +\ k$. 


The divisor is $3x^2\ -\ 5$. 

To do:


We have to find the polynomial that must be added to the polynomial  $f(x)\ =\ x^4\ +\ 2x^3\ -\ 2x^2\ +\ x\ -\ 1$  so that the resulting polynomial is exactly divisible by  $x^2\ +\ 2x\ -\ 3$.


Solution:


Let the remainder when $x^2\ +\ 2x\ -\ 3$ divides $f(x)\ =\ x^4\ +\ 2x^3\ -\ 2x^2\ +\ x\ -\ 1$ be $r(x)$.

Therefore,

Dividend$=x^4+2x^3-2x^2+x-1$

Divisor$=x^2+2x-3$

 $x^2+2x-3$)$x^4+2x^3-2x^2+x-1$($x^2+1-15$

                       $x^4+2x^3-3x^2$

               -------------------------------


                                        $x^2+x-1$

                                          $x^2+2x-3$            

                                         -------------


                                              $-x+2$ 

Remainder$r(x)=-x+2$

If we subtract the remainder from the dividend then it is completely divisible by the divisor.

 Therefore, we must add $-r(x)=-(-x+2)=x-2$.

 The polynomial that must be added to the polynomial  $f(x)\ =\ x^4\ +\ 2x^3\ -\ 2x^2\ +\ x\ -\ 1$  so that the resulting polynomial is exactly divisible by  $x^2\ +\ 2x\ -\ 3$ is $x-2$.

Tutorialspoint
Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

36 Views

Kickstart Your Career

Get certified by completing the course

Get Started
Advertisements