Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:
(i) $p(x) = x^3 - 3x^2 + 5x -3, g(x) = x^2-2$
(ii) $p(x) =x^4 - 3x^2 + 4x + 5, g(x) = x^2 + 1 -x$
(iii) $p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$

AcademicMathematicsNCERTClass 10

To do: 

We have to divide the polynomial $p( x)$ by the polynomial $g( x)$ and find the quotient and remainder in each case.

Solution:

(i) As given, $( p(x)=x^{3}-3 x^{2}+5 x-3$, $g(x)=x^{2}-2$

On Dividing $p( x)$ by $g( x)$ by long division:

$x^2-2$)$x^3-3x^2+5x-3$($x-3$

                $x^3-2x$

            ---------------------

                  $-3x^2+7x-3$

                  $-3x^2+6$

              --------------------

                           $7x-9$

Quotient$=x-3$

Remainder$=7x-9$.

(ii) $p(x) = x^4 - 3x^2 + 4x + 5$

$g(x) = x^2+1 -x$

$x^2-x+1$)$x^4-3x^2+4x+5$($x^2+x-3$

                    $x^4+x^2-x^3$

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

                             $x^3-4x^2+4x+5$

                            $x^3-x^2+x$

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

                                   $-3x^2+3x+5$

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

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

                                                    $8$

                                                ---------

Therefore, the quotient is $x^2+x-3$ and the remainder is $8$.

(iii) $p(x) = x^4 - 5x + 6$

$g(x) = 2 -x^2$

$2-x^2$)$x^4-5x+6$($-x^2-2$

               $x^4-2x^2$

            ------------------

               $2x^2-5x+6$

               $2x^2-4$

          --------------------

                       $-5x+10$

Therefore, the quotient is $-x^2-2$ and the remainder is $-5x+10$.

raja
Updated on 10-Oct-2022 13:19:34

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