Question

on dividing x4 - 2x3 + 3x2 - ax =b by ( x - 1) and ( x + 1) we get remainder 5 and 19 respectively find the remainder when f (x) is divided by (x-2)

Answer 1

put x=1
$1 - 2 + 3 - a - b = 5 \\ - a - b = 2$
put x = -1
$1 + 2 + 3 + a - b = 19 \\ a - b = 13$
from these two equations solve the value of a and b.
add both equations
$- 2b = 15 \\ b = \frac{ - 15}{2}$

$a - ( \frac{ - 15}{2} ) = 13 \\ a = 13 - \frac{15}{2} = \frac{26 - 15}{2} = \frac{11}{2}$
put the value of a and b in polynomial
${x}^{4 } - 2 {x}^{3} + 3 {x}^{2} - \frac{11}{2} x - \frac{ - 15}{2}$
put x=2 in the equation,to find the value of remainder
=16-16+27-11+15
=31 is the answer