(Q6)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:

Given polynomial x4-6x3-16x2+25x+10 is divided by another polynomial x2-2x+k

Remainder x+a

Let us divide

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

2-x-(+)
x2-2x+k) 4-x3-x2+x+
4 - x3 + x2
-   +   -
-x3-2(+)+x
-x3+x2      -kx
+   -        +
-2(k+)+x(+k)+
-2(k+)+2x(k+)-k(+)
+     -         +
x(k+-k-)++k(+)

Remainder

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

= x(2k-23)+(k2+24k+10)

Give remainder x+a

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

2k - 23= ----------(1)

2k=+23=

k=
2
 =

k2+24k+10 = a

Subsitituting 'k' value

()2+24()+10=a

a=

Required k= and 'a' =