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On dividing the polynomial $f x = x 3 + 2 x 2 − 5 x − 6$ by another polynomial $g (x),$ the quotient and the remainder come out be respectively $x + 1$ and $− 4 x − 4 .$ What will be the polynomial $g (x)$ ?

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x 2 − x − 2

x 2 + x − 2

x 2 + x + 2

x 2 − x + 2

answer is B.

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Detailed Solution

Given,

f x = x 3 + 2 x 2 − 5 x − 6 q x = x + 1 r x = − 4 x − 4

By Division Algorithm, we have,

f x = g x × q x + r x ⇒ g (x) = f (x) − r (x) q (x) (1)

Substituting the values, we get

g (x) = x 3 + 2 x 2 − 5 x − 6 − − 4 x − 4 x + 1 ⇒ g (x) = x 3 + 2 x 2 − x − 2 x + 1

We get,

⇒ g (x) = x 2 + x − 2

Therefore on dividing the polynomial

f x = x 3 + 2 x 2 − 5 x − 6

by another polynomial

g (x) = x 2 + x − 2,

the quotient and the remainder come out be respectively

x + 1

and

− 4 x − 4 .

Hence option (2) is correct.

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