Question

If f(x)=x^3+lx+m leaves the same reminder when divided by (x-1) or by (x+1). Find values of l and m?

Answer 1

When f(x) is divided by (x+1) and (x-1) , the remainders are 19 and 5 respectively . 

∴ f(-1) = 19 and f(1) = 5

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

⇒ 1 +2 + 3 + a + b = 19

∴ a + b = 13 ------- (i)

Again , f(1) = 5

⇒ 14 - 2 × 13 + 3 × 12 - a × 1 b = 5

⇒ 1 - 2 + 3 - a + b = 5

∴ b - a = 3 ------ (ii)

solving eqn (i) and (ii) , we get 

a = 5 and b = 8

Now substituting the values of a and b in f(x) , we get 

 ∴ f(x) = x4 - 2x3 + 3x2 - 5x + 8

Now f(x) is divided by (x-3) so remainder will be f(3) 

 ∴ f(x) =  ∴ f(x) = x4 - 2x3 + 3x2 - 5x + 8

⇒ f(3) = 34 - 2 × 33 + 3 × 32 - 5 × 3 + 8 

= 81 - 54 + 27 - 15 + 8 = 47 

Answer 2

Answer:

l= -1 and m=7 is the answer.