Question

Anyone has the power to solve sum no-11, if yes so solve

Answer 1

Question :-

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

Solution :-

Given that when f(x) is divided by (x - 1) the remainder is 5. So by remainder theorem when x = 1, the remainder is 5

=> x⁴ - 2x³ + 3x² - ax + b = 5 at x = 1

=> (1)⁴ - 2(1)³ + 3(1)² - a(1) + b = 5

=> 1 - 2 + 3 - a + b = 5

=> 2 - a + b = 5

=> - a + b = 5 - 2

=> b - a = 3........... ( 1 )

Now if f(x) is divided by (x + 1), the remainder is 19.

=> x⁴ - 2x³ + 3x² - ax + b = 19 at x = - 1

=> (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + b = 19

=> 1 - 2(-1) + 3(1) + a + b = 19

=> 1 + 2 + 3 + a + b = 19

=> 6 + a + b = 19

=> a + b = 19 - 6

=> a + b = 13 ................. ( 2)

Now add ( 1 ) and (2)

=> b - a + a + b = 3 + 13

=> 2b = 16

=> b = 16/2

=> b = 8

So b - a = 3

=> 8 - a = 3

=>. - a = 3 - 8

=> - a = - 5

=> a = 5

Values of a and b is 5 and 8 respectively.

Now to find the remainder when f(x) is divided by (x - 2)

So to find the remainder, put the value of x as 2

=> (2)⁴ - 2(2)³ + 3(2)² - a(2) + b

=> 16 - 2(8) + 3(4) - 5(2) + 8

=> 16 - 16 + 12 - 10 + 8

=> 2 + 8

= 10

The remainder is 10