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# 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):

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