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) + 0.. (a) is not
true
(b) Ifp(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.
... (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) 0.
.. r(x)=g(x) is not true.

Solution

(a, b, d)
(a) If p(x) is not divisible by g(x), then r(x) + 0.. (a) is not
true
(b) Ifp(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.
... (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) 0.
.. r(x)=g(x) is not true.