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 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 *x*^{2} – 2*x* + *k* leaves (*x *+ *a*) as remainder.

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

⇒ (–10 + 2*k*)*x* + (10 – *a* – 8*k* + *k*^{2}) = 0

⇒ –10 + 2*k* = 0 and 10 – *a* – 8*k* + *k*^{2 }= 0

⇒ *k *= 5 and 10 – *a* – 8 (5) + 5^{2} = 0

⇒ *k *= 5 and – *a* – 5 = 0

⇒ *k *= 5 and *a* = –5

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