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The remainder on division of $x 3 + 2 x 2 + k x + 3$ by $x − 3$ is $21.$ Find the quotient and the value of $k$ .

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k = 7

and quotient

= 2 x 2 − 7 x + 14

k = − 9

and quotient

= x 2 + 5 x + 6

k = 9

and quotient

= x 2 − 5 x − 6

k = − 7

and quotient

= 2 x 2 + 7 x − 14

answer is A.

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Detailed Solution

Given that when the polynomial

x 3 + 2 x 2 + k x + 3

is divided by

x − 3

we get the remainder as

21.

We need to find the quotient and value of

k .

We will use the long division method for performing the division.
We have,
Question Image

Quotient is

x 2 + 5 x + (15 + k) ... (i)

Remainder is

3 + 3 (15 + k)

. But from the question we know that remainder is

21.

Therefore, we have,

21 = 3 + 3 (15 + k) ⇒ 21 = 3 + 75 + 3 k ⇒ 18 = 45 + 3 k ⇒ − 27 = 3 k ⇒ k = − 27 3 ∴ k = − 9

Equating the value of

k

in equation

i

, we have,

x 2 + 5 x + (15 + k) = x 2 + 5 x + (15 − 9) ⇒ x 2 + 5 x + (15 + k) = x 2 + 5 x + 6

Hence, the value of

k

− 9

and the quotient is

x 2 + 5 x + 6

.
Therefore, option 1 is correct.

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The remainder on division of x 3 +2 x 2 +kx+3 by x−3 is 21. Find the quotient and the value of k .

Detailed Solution

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The remainder on division of $x 3 + 2 x 2 + k x + 3$ by $x − 3$ is $21.$ Find the quotient and the value of $k$ .