Divide the polynomial $p (x)$ by the polynomial $g (x)$ and find the quotient $q (x)$ and reminder $r (x)$ in each case:
$p (x) = x^{4} + 1, g (x) = x + 1$

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Solution

$p (x) = x^{4} + 1, g (x) = x + 1$
Dividend $= x^{4} + 1$
Divisor $= x + 1$
$q (x) = x^{3} - x^{2} + x - 1$
$r (x) = 0$

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On dividing a polynomial p(x) by a nonzero polynomial q(x), let g(x) be the quotient and r(x) be the remainder then p(x) = q(x).g(x)+r(x), where

(a) r(x) = 0 always

(b) deg r(x) < deg g(x) always

(d) r(x) = g(x)

Q. Divide the polynomial

p (x)

by the polynomial

g (x)

and find the quotient and remainder.

p (x) = x^{4} - 3 x^{2} + 4 x + 5

g (x) = x^{2} + 1 - x

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

Q. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder of the following.

p (x) = x^{4} - 3 x^{2} + 4 x + 5

g (x) = x^{2} + 1 - x

Q. A polynomial

p (x)

is divided by

g (x)

, the obtained quotient

q (x)

and the remainder

r (x)

are given in the table. Find

p (x)

in each case.

Sl.	$p (x)$	$g (x)$	$q (x)$	$r (x)$
i	?	$x - 2$	$x^{2} - x + 1$	$4$
ii	?	$x + 3$	$2 x^{2} + x + 5$	$3 x + 1$
iii	?	$2 x + 1$	$x^{3} + 3 x^{2} - x + 1$	$0$
iv	?	$x - 1$	$x^{3} - x^{2} - x - 1$	$2 x - 4$
v	?	$x^{2} + 2 x + 1$	$x^{4} - 2 x^{2} + 5 x - 7$	$4 x + 12$

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Divide the polynomial p(x) by the polynomial g(x) and find the quotient q(x) and reminder r(x) in each case:p(x)=x4+1,g(x)=x+1

p(x)=x4+1,g(x)=x+1Dividend =x4+1Divisor =x+1q(x)=x3−x2+x−1r(x)=0

Divide the polynomial $p (x)$ by the polynomial $g (x)$ and find the quotient $q (x)$ and reminder $r (x)$ in each case:
$p (x) = x^{4} + 1, g (x) = x + 1$

$p (x) = x^{4} + 1, g (x) = x + 1$
Dividend $= x^{4} + 1$
Divisor $= x + 1$
$q (x) = x^{3} - x^{2} + x - 1$
$r (x) = 0$