The polynomial f(x) =2x+3x -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).
Answers
Answer :
Given :
Expression is f(x) = 2x + 3x - ax + b
(x-1) when divided leaves remainder 5
(x+1) when divided leaves remainder 19
Required to find :
- Values of " a " and " b "
Solution :
Let's consider the given expression
f(x) = 2x + 3x - ax + b
(x-1) when divided leaves remainder 5
So, let
x - 1 = 0
x = 1
Hence,
f (1) = 2(1) + 3(1) - a(1) + b = 5
2 + 3 - a + b - 5 = 0
5 - a + b - 5 = 0
0 - a + b = 0
Then ,
b = a. -----------> equation 1
Similarly,
(x+1) when divided leaves remainder 19
So,
Let x + 1 = 0
x = - 1
Hence,
f(- 1) = 2(-1) + 3(-1) - a (-1) + b = 19
- 2 - 3 + a + b = 19
- 5 + a + b - 19 = 0 ( substitute the value of " b " from equation 1 )
- 5 + a + a - 19 = 0
- 24 + 2a = 0
2a = 24
a = 24/2
a = 12
hence ,
Value of a = 12
value of b = ?
but b = a ( from equation 1 )
So, b = 12
Therefore,
Value of a = 12
Value of b = 12
This states that ,
a = b = 12