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What must be added to the polynomial $f x = x 4 + 2 x 2 − 2 x 2 + x − 1$ so that the resulting polynomial is exactly divisible by $x 2 + 2 x − 3 ?$

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x + 2

− x + 2

− x − 2

x − 2

answer is B.

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Detailed Solution

Given polynomial

f x = x 4 + 2 x 2 − 2 x 2 + x − 1

and divisor

g (x) = x 2 + 2 x − 3.

Division algorithm is given by,

,where , f(x), g(x), q(x) and r(x) are dividend, functions, quotient and remainder respectively.
Divide the polynomial

f x = x 4 + 2 x 3 − 2 x 2 + x − 1

g x = x 2 + 2 x − 3.

∴ r x = − x + 2

Now, check the polynomial,
Question Image

Thus,

− x + 2

must be added to the polynomial f(x) so that it gets completely divisible by g(x).
Therefore the remainder

− x + 2

must be added to the polynomial

f x = x 4 + 2 x 2 − 2 x 2 + x − 1

so that the resulting polynomial is exactly divisible by

x 2 + 2 x − 3.

Hence the correct option is 2.

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