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Division of a Polynomial by a Monomial

If fx= 2 x ...

Question

If $f (x) = 2 x^{4} - 5 x^{3} + x^{2} + 3 x - 2$ is divisible by $g (x)$ the quotient is $q (x) = 2 x^{2} - 5 x + 3$ and $r (x) = - 2 x + 1$ find $g (x)$ .

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Solution

$f (x) = g (x) q (x) + r (x)$
$2 x^{4} - 5 x^{3} + x^{2} + 3 x - 2 = (g (x)) (2 x^{2} - 5 x + 3) + (- 2 x + 1)$
$g (x) (2 x^{2} - 5 x + 3) = 2 x^{4} - 5 x^{3} + x^{2} + 5 x - 3$
$g (x) = \frac{2 x^{4} - 5 x^{3} + x^{2} + 5 x - 3}{2 x^{2} - 5 x + 3}$
(Image )
$∴ g (x) = x^{2} - 1$

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Similar questions

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

is divided by g(x) the qoutient is

q (x) = 2 x^{2} - 5 x + 3 a n d r (x) = - 2 x + 1

Q. Apply division algorithm to find the quotient q(x) and remainder r(x) in dividing f(x) by g(x) in each of the following :

(i) f(x) = x³ − 6x² + 11x − 6, g(x) = x² + x + 1
(ii) f(x) = 10x⁴ + 17x³ − 62x² + 30x − 3, g(x) = 2x² + 7x + 1
(iii) f(x) = 4x³+ 8x + 8x² + 7, g(x) = 2x² − x + 1
(iv) f(x) = 15x³ − 20x² + 13x − 12, g(x) = 2 − 2x + x²

Q. Find the quotient

q (x)

r (x)

of the following when

f (x)

g (x)

.

f (x) = x^{4} + 2 x^{2} + 1; g (x) = x^{3} + 1

Q. A polynomial

p (x)

g (x)

, the obtained quotient

q (x)

and the remainder

r (x)

are given in the table. Find

p (x)

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

Q. Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing

f (x) = 10 x^{4} + 17 x^{3} - 62 x^{2} + 30 x - 3

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

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