# the polynomial f(x)=x4-2x3+3x2-ax+b when divided by (x-1) and (x+1) 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). find the values of a and b so that the polynomial (9x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2).

1.

the given polynomial is

when f(x) is divided by (x-1) remainder is 5.

x-1=0 ⇒ x =1

therefore f(1) = 5

when f(x) is divided by (x+1) remainder is 19.

therefore x+1 = 0  ⇒ x = -1

therefore f( -1) = 19

adding eq(1) and eq(2):

put b = 8 in eq(1):

thus the values of a is 5 and b is 8.

thus the given polynomial is

if f(x) is divided by x-2 , put x-2 = 0 ⇒ x = 2

therefore remainder = f(2)

2.

let the given polynomial is

if f(x) is exactly divisible by (x-1) as well as (x-2).

therefore the remainder , when dividing by (x-1) and (x-2) is 0.

put x-1 = 0 ⇒ x=1 and x-2 = 0 ⇒ x=2

therefore f(1)=0 and f(2) =0

subtracting eq(2) by eq(1):

put a = -33 in eq(1):

thus the values of a and b are -33 and 34 respectively.

hope this helps you.

cheers!!

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