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Q.

What are conditions required for r(x) to divide a polynomial p(x) by a non-zero polynomial q(x), where g(x) is the quotient and r(x) is the remainder and $p (x) = q (x) ⋅ g (x) + r (x)$ .

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a

r (x) = 0

b

deg r (x) < d e g g (x)

c

r (x) = 0 o r d e g r (x) < deg g (x)

d

r (x) = g (x)

answer is C.

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Detailed Solution

It is given that on dividing a polynomial p(x) by a non-zero polynomial q(x), let g(x) be the quotient and r(x) be the remainder, then

p (x) = q (x) ⋅ g (x) + r (x)

.
Now, whenever a polynomial is divided by another polynomial of some degree, then either a remainder is obtained or the polynomial is exactly divisible.
In the case when the remainder is obtained, then the degree of the remainder is always less than that of the polynomial which is the divisor.
So, if we assume the remainder to be

r (x)

, and if the dividend polynomial is exactly divisible, then

r (x) = 0

. And if there is some remainder obtained, then

deg r (x) < deg g (x)

, where

g (x)

is the divisor polynomial.
So, the conditions for

p (x) = q (x) ⋅ g (x) + r (x)

are that

r (x) = 0 o r d e g r (x) < deg g (x)

.
Hence, the correct option is (3).

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