when u divide (x^4-6x^3+16x^2-25x+10) by (x^2-2x+k) , the remainder is x+a find a and k

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Blazing goIITian

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Here is a way to cut down the work..

x^2 equiv (2x-k) mod (x^2 -2x+k)

herefore x^4 - 6x^3+16x^2-25x+10 equiv (2x-k)^2-6(2x^2-kx)+16(2x-k)-25x+10

equiv -8x^2+x(2x+7)+(k^2-16k+10) equiv 8(k-2x)+x(2k+7)+(k^2-16k+10)

equiv x(2k-9) + (k^2-8k+10)

Now equate the co-efficiets ,

$2k-9=1\\Rightarrow k=5$

Also,

$k^2-8k+10=a\\Rightarrow a=-5$

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