Question

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.

Answer 1

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 – 2x + k  leaves (x + a) as remainder.

⇒ (4k – 25 + 16 – 2k)x + [10 – k(8 – k) ] = x + a  
⇒ (2k – 9)x + [10 – 8k + k² ] = x + a
On comparing both the sides, we get
2k  – 9 = 1
⇒ 2k = 10
∴ k = 5
Also 10 – 8k + k² = a
⇒ 10 – 8(5) + 5² = a
⇒ 10 – 40 + 25 = a
∴ a = – 5