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) $p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
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 $-5x+10$.
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