Question # Which of the following given options is/are incorrect?
If p(x)=9(x) g(x) +r(x) (By Division Algorithm) where p(x),
g(x) are any two polynomials with g(x) + 0, then
(a) r(x)=0 always
(b) degree of r(x) < degree of g(x) always
(c) either r(x)= 0 or degree of r(x)

# Which of the following given options is/are incorrect?
If p(x)=9(x) g(x) +r(x) (By Division Algorithm) where p(x),
g(x) are any two polynomials with g(x) + 0, then
(a) r(x)=0 always
(b) degree of r(x) < degree of g(x) always
(c) either r(x)= 0 or degree of r(x)

Solution ( (a, b, d) )
(a) If ( p(x) ) is not divisible by ( g(x), ) then ( r(x) eq 0 therefore(a) ) is not
true
(b) If ( p(x) ) is divisible by ( g(x) ), then ( r(x)=0 ) for all ( x ) i.e., ( r(x) ) is a zero polynomial whose degree is not defined. ( therefore(b) ) is not true
(c) is clearly true [ : division algorithm rule]
(d) since degree of ( r(x)< ) degree of ( g(x) ) or ( r(x)=0, ) but ( g(x) eq 0 )
( therefore r(x)=g(x) ) is not true.

Solution