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 ? $x^{2} + 2 x + 1$ $x^{4} - 2 x^{2} + 5 x - 7$ $4 x + 12$

Open in App

Solution

A polynomial $p (x)$ is defined as
$\Rightarrow p (x) = g (x) q (x) + r (x)$

where $g (x) =$ divisor ; $q (x) =$ quotient and $r (x) =$ remainder

$∴$ $p (x)$ can be found by multiplying $g (x)$ with $q (x)$ $&$ adding $r (x)$ to the product.

$(i) . g (x) = (x - 2);$ $q (x) = x^{2} - x + 1;$ $r (x) = 4$
$∴ p (x) = (x - 2) [x^{2} - x + 1] + 4$
$= x^{3} - x^{2} + x - 2 x^{2} + 2 x - 2 + 4$
$= x^{3} - 3 x^{2} + 3 x + 2$

$(i i) . g (x) = (x + 3);$ $q (x) = 2 x^{2} + x + 5;$ $r (x) = 3 x + 1$
$∴ p (x) = (x + 3) [2 x^{2} + x + 5] + (3 x + 1)$
$= 2 x^{3} + x^{2} + 5 x + 6 x^{2} + 3 x + 15 + 3 x + 1$
$= 2 x^{3} + 7 x^{2} + 11 x + 16$

$(i i i) . g (x) = (2 x + 1);$ $q (x) = x^{3} + 3 x^{2} - x + 1;$ $r (x) = 0$
$∴ p (x) = (2 x + 1) [x^{3} + 3 x^{2} - x + 1] + (0)$
$= 2 x^{4} + 6 x^{3} - 2 x^{2} + 2 x + x^{3} + 3 x^{2} - x + 1$
$= 2 x^{4} + 7 x^{3} + x^{2} + x + 1$

$(i v) . g (x) = (x - 1);$ $q (x) = x^{3} - x^{2} - x - 1;$ $r (x) = 2 x - 4$
$∴ p (x) = (x - 1) [x^{3} - x^{2} - x - 1] + (2 x - 4)$
$= x^{4} - x^{3} - x^{2} - x - x^{3} + x^{2} + x + 1 + 2 x - 4$
$= x^{4} - 2 x^{3} + 2 x - 3$

$(v) . g (x) = (x^{2} + 2 x + 1);$ $q (x) = x^{4} - 2 x^{2} + 5 x - 7;$ $r (x) = 4 x + 12$
$∴ p (x) = (x^{2} + 2 x + 1) [x^{4} - 2 x^{2} + 5 x - 7] + (4 x + 12)$
$= x^{6} - 2 x^{4} + 5 x^{3} - 7 x^{2} + 2 x^{5} + 4 x^{3} + 10 x^{2} - 14 x + x^{4} - 2 x^{2} + 5 x - 7 + 4 x + 12$
$= x^{6} - x^{4} + x^{3} + x^{2} + 2 x^{5} - 5 x + 5$
$= x^{6} + 2 x^{5} - x^{4} + x^{3} + x^{2} - 5 x + 5$
Hence, solve.

Suggest Corrections

Similar questions

Now in each of the following, find the quotient and remainder on dividing p(x) by q(x) and write then in the form p(x) = u(x) q(x) + v(x).

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

On dividing f(x) = x³ - 3x² + x + 2 by a polynomial g(x) the quotient and remainder q(x) and r(x) are (x – 2) and (-2x + 4) respectively, then g(x) is

Q. 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} - 3 x^{2} + 5 x - 3, g (x) = x^{2} - 2

(ii)

p (x) = x^{4} - 3 x^{2} + 4 x + 5, g (x) = x^{2} + 1 - x

(iii)

p (x) = x^{4} - 5 x + 6, g (x) = 2 - x^{2}

Q. Find the quotient

q (x)

and remainder

r (x)

of the following when

f (x)

is divided by

g (x)

p (x) = x^{6} + x^{4} - x^{2} - 1

;

g (x) = x^{3} - x^{2} + x - 1

If dividend = $x^{4} + x^{3} - 2 x^{2} + x + 1$ and divisor = $(x - 1)$ , find the quotient, $q (x)$ and remainder $r (x)$ .

Join BYJU'S Learning Program

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	?	$x^{2} + 2 x + 1$	$x^{4} - 2 x^{2} + 5 x - 7$	$4 x + 12$

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−2x2−x+14ii?x+32x2+x+53x+1iii?2x+1x3+3x2−x+10iv?x−1x3−x2−x−12x−4v?x2+2x+1x4−2x2+5x−74x+12