If the polynomial x⁴ - 6x³ + 16x² - 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a, find the values of k and a.

We know that,

Dividend = Divisor × Quotient + Remainder

⇒ Dividend – Remainder  =  Divisor × Quotient

⇒ Dividend – Remainder is always divisible by the divisor.

Now, it is given that  f(x)  when divided by  x² – 2x + k  leaves (x + a) as remainder.

So,  for  f(x) to be completely divisible by  x² – 2x + k,  remainder must be equal to zero

⇒  (–10 + 2k)x + (10 – a – 8k + k²) = 0

⇒  –10 + 2k = 0  and  10 – a – 8k + k² =  0

⇒  k = 5  and  10 – a – 8 (5) + 5² = 0

⇒  k = 5  and   – a – 5 = 0

⇒  k = 5  and   a  =  –5