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 x + a, find x and a.

Solution :

Let us divide \(x^4 – 6x^3 + 16x^2 – 25x + 10\) by \(x^2 – 2x + k\)

polynomial image

\(\therefore\)  Remainder = (2k – 9)x – (8 – k)k + 10

But the remainder is given as x + a,

On comparing their coefficients, we have :

2k – 9 = 1  \(\implies\)  k = 5

and -(8 – k)k + 10 = a

So, a = -(8 – 5)5 + 10

a = -15 + 10 = – 5

Hence, k = 5 and a = -5

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