Question

what must be added to 6x^5+5x^4+11x^3-3x^2+x+5 so that it may be exactly divisible by 3x^2-2x+4

Answer 1

GIVEN :

The polynomial $6x^5+5x^4+11x^3-3x^2+x+5$ is divisible by $3x^2-2x+4$

TO FIND :

The remainder of $6x^5+5x^4+11x^3-3x^2+x+5$ is divisible by $3x^2-2x+4$

SOLUTION :

Given that the polynomial is $6x^5+5x^4+11x^3-3x^2+x+5$ is divisible by [tex]3x^2-2x +4 [/tex]

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

                                     ________________________

               $3x^2-2x +4$ ) $6x^5+5x^4+11x^3-3x^2+x+5$

                                      $6x^5- 4x^4+ 8x^3$    (-)

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

                                               $9x^4+3x^3-3x^2$

                                               $9x^4-6x^3+12x^2$   (-)

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

                                                    $9x^3-15x^2+ x$

                                                    $9x^3- 6x^2+12x$  (-)

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

                                                           $-9x^2- 11x + 5$

                                                           $-9x^2+ 6x -12$  (-)

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

                                                                      -17x + 17

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

∴ Additive inverse of -17x + 17 is 17x - 17

Hence 17x - 17 is added to $6x^5+5x^4+11x^3-3x^2+x+5$,so that the polynomial so obtained is exactly divisible by $3x^2-2x +4$

Answer 2

Answer:

GIVEN :

The polynomial 6x^5+5x^4+11x^3-3x^2+x+56x

5

+5x

4

+11x

3

−3x

2

+x+5 is divisible by 3x^2-2x+43x

2

−2x+4

TO FIND :

The remainder of 6x^5+5x^4+11x^3-3x^2+x+56x

5

+5x

4

+11x

3

−3x

2

+x+5 is divisible by 3x^2-2x+43x

2

−2x+4

SOLUTION :

Given that the polynomial is 6x^5+5x^4+11x^3-3x^2+x+56x

5

+5x

4

+11x

3

−3x

2

+x+5 is divisible by 3x^2-2x +43x

2

−2x+4

2x^3+3x^2+3x-32x

3

+3x

2

+3x−3

________________________

3x^2-2x +43x

2

−2x+4 ) 6x^5+5x^4+11x^3-3x^2+x+56x

5

+5x

4

+11x

3

−3x

2

+x+5

6x^5- 4x^4+ 8x^36x

5

−4x

4

+8x

3

(-)

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

9x^4+3x^3-3x^29x

4

+3x

3

−3x

2

9x^4-6x^3+12x^29x

4

−6x

3

+12x

2

(-)

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

9x^3-15x^2+ x9x

3

−15x

2

+x

9x^3- 6x^2+12x9x

3

−6x

2

+12x (-)

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

-9x^2- 11x + 5−9x

2

−11x+5

-9x^2+ 6x -12−9x

2

+6x−12 (-)

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

-17x + 17

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

∴ Additive inverse of -17x + 17 is 17x - 17

Hence 17x - 17 is added to 6x^5+5x^4+11x^3-3x^2+x+56x

5

+5x

4

+11x

3

−3x

2

+x+5 ,so that the polynomial so obtained is exactly divisible by 3x^2-2x +43x

2

−2x+4

Step-by-step explanation:

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