If the polynomial \(x^4- 6x^3 + 16x^2 -25x + 10\) is divided by another polynomial \(x^2 -2x + k \) the remainder comes out to be \(x +a\) find \(k\) and \(a\).

Solving it using long division:

\(\begin{array}{rrrr|ll} x^4 -6x^3 & +16x^2 & -25x & +10 & x^2 -2x + k\\ -x^4 + 2x^3 & -kx^2 & & & x^2 -4x + (8-k) \\ \hline -4x^3 & +(16-k)x^2 & -25x & +10 \\ \phantom{-}+4x^3 & -8x^2 & + 4kx & & & \\ \hline & (8-k)x^2 & +(4k-25)x & +10 \\ & -(8-k)x^2 & +2(8-k)x & -(8-k)k \\ \hline & (2k-9)x & -(8-k)k & +10\\ \hline \end{array}\)

Here remainder = \((2k-9)x -(8-k)k +10\)...........(i)

Given remainder = \(x + a\)..........................(ii)

On comparing the coefficients x in equation (i) and (ii):

=>\(2k - 9 = 1\)
=>\(2k = 10\)
=>\(k = 5\)

On comparing the a in equation (i) and (ii):

=>\(-(8-k)k + 10 = a\)
=>\(-(8-5)5 + 10 = a\)
=>\(a = -15 + 10 = -5\)