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The quotient and remainder when the polynomial $p (x) = 3 x 3 + 4 x 2 + 5 x − 13$ is divided by a polynomial $g (x)$ are $3 x + 10$ and $16 x − 43$ respectively, then what will be $g (x)$ ?

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− x 2 − 2 x + 3

x 2 − 2 x + 3

x 2 + 2 x + 3

− x 2 + 2 x + 3

answer is B.

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Detailed Solution

Here, we are given

p x = 3 x 3 + 4 x 2 + 5 x − 13 g x = ? q x = 3 x + 10 r x = 16 x − 43

By Division Algorithm, we have,

p x = g x q x + r x (1)

Substituting the values, we get,

p x = g x × q x + r x ⇒ 3 x 3 + 4 x 2 + 5 x − 13 = g (x) 3 x + 10 + 16 x − 43 ⇒ 3 x 3 + 4 x 2 − 11 x + 30 = g (x) 3 x + 10 ⇒ g x = 3 x 3 + 4 x 2 − 11 x + 30 3 x + 10

We have,

⇒ g x = x 2 − 2 x + 3 3 x + 10 3 x + 10 ⇒ g x = x 2 − 2 x + 3

Therefore the value of

g x = x 2 − 2 x + 3 .

Hence, option (2) is correct.

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