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The polynomials $a x 3 + 3 x 2 − 3$ and $2 x 3 − 5 x + a$ when divided by $(x − 4)$ leave the same remainder. The value of the remainder is,

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− 1

1

0

a

answer is A.

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Detailed Solution

Let us assume that

f x = a x 3 + 3 x 2 − 3

and

g x = 2 x 3 − 5 x + a

.
According to the given information,

f x

and

g x

give the same remainders when divided by

(x − 4)

, that is

f (4)

and

g (4)

are the remainders of both the polynomials when they are divided by

(x − 4)

, by Remainder Theorem.
Hence,

⇒ f (4) = g (4) ⇒ a (4) 3 + 3 (4) 2 − 3 = 2 (4) 3 − 5 (4) + a ⇒ 64 a + 48 − 3 = (64 × 2) − 20 + a ⇒ 64 a + 45 = 128 − 20 + a ⇒ 64 a − a = 108 − 45 ⇒ 63 a = − 63 ⇒ a = − 1

Therefore, the required remainder is -1.
Hence, option 1 is correct.

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