Question

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 k and a.

Answer 1

Given polynomial x⁴ – 6x³ – 16x² + 25x + 10 and another polynomial is x² – 2x + k. 

Remainder is x + a 

Let us divide

x⁴ – 6x³ – 16x² + 25x + 10 by x² – 2x + k

∴ Remainder 

= x(4k + 25 – 2k – 48) + 10 + k(k + 24) 

= x(2k – 23) + (k² + 24k + 10) 

Given remainder is x + a 

on comparing the coefficients of x and constant terms on both sides 

2k – 23 = 1 ……. (1) 

2k = 1 + 23 = 24

⇒ k = 24/2 = 12 

k² + 24k + 10 = a …….. (2) 

Substitute ‘k’ value in equation (2) 

(12)² + 24(12) + 10 = a 

144 + 288 + 10 = a 

⇒ a = 442 

∴ Required k = 12 and ‘a’ = 442