Question

the polynomial f(x) = x4-2x3+3x2-ax+b when divided by x-1 and x+2 leaves the remainders 5 and 19 respectively . find the values of a and b . hence find the remainder when f(x)is divided by x-2.

Answer 1

as given remainder is 19

So, b + 2a + 44 = 19


b + 2a = 19 - 44

b + 2a = -25 ........ ( ii )

Sub ( ii ) from ( i )


b - a - ( b + 2a ) = 3 - ( -25 )

b - a - b - 2a = 3 + 25

-3a = 28

a = - 28/3

Putting the value in equation ( i )

b - a = 3

b + 28/3 = 3

( 3b + 28 )/3 = 3

3b + 28 = 9

3b = 9 - 28

3b = -19

b = -19/3

When we put the value in equation
we get ,

3x⁴ - 6x³ + 9x² + 28x -19 = 0

Dividing this by x - 2

x-2 ) 3x⁴-6x³+9x²+28x-19(

Then,

quotient come = 3x³+9x+46

and remainder = 73


@Altaf