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If the polynomial $x 4 − 6 x 3 + 16 x 2 − 25 x + 10$ is divided by another polynomial $x 2 − 2 x + k$ , the remainder comes out to be $x + a,$ the values of $k$ and $a$ are:

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k= 5, a = -5

k= -5, a = -5

k= -5, a = 5

k= 5, a = 5

answer is A.

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Detailed Solution

Given a polynomial

x 4 − 6 x 3 + 16 x 2 − 25 x + 10

is divided by another polynomial

x 2 − 2 x + k

, the remainder comes out to be

x + a

.
Dividing the polynomial by

x 2 − 2 x + k

Remainder

= (2 k − 9) x + 10 − k (8 − k)

But remainder = x + a
So, equating both and comparing,

(2 k − 9) x + 10 − k (8 − k) = x + a

Comparing x,

2 k − 9 = 1

2 k = 1 + 9

2 k = 10

k = 5

Comparing constant,

10 − k (8 − k) = a

10 − 8 k + k 2 = a

10 − (8 × 5) + 52 = a

10 − 40 + 25 = a

a = − 5

Hence, option 1 is correct.

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