急!!!!!F4 maths Polynomial- Remainder Theorem&Factor Theorem

a ) When f ( x ) is divided by x + 1 , the remainder is 1 . Find the value of k


Remainder = f(-1) = -1 ^99 + k
Since Remainder = 1,
1 = -1 ^99 + k
1 = -1 + k
k=2

b ) Hence, find the remainder when 9 ^ 99 is divided by 10

let g(x) = f(x) -2 = x^99
when g(x) is divided by (x+1),
remainder = g(-1) = f(-1) -2 = 1 - 2 = -1
Put x =9, 9^99 is divided by 10 which gives remainder -1.
We can also say 9^99 is divided by 10 which gives remainder 10-1 = 9.
For example,
79 is divided by 10. Remainder is -1 because 79 - 8*10 = -1.
But 79 - 7*10 = 79 - 70 = 9. So, remainder 9 is equivalent to remainder -1.

let g(x) be a function
f(x) = g(x) (x+1) + 1
f(9) = g(9)(9+1) +1
9^99 + 2 = g(9) (10) + 1
9^99 = g(9) (10) -1
9^99 = [g(9)-1] (10) + 10 -1
9^99 = [g(9)-1] (10) + 9

therefore, the remainder is 9

From a
when n=9 ,9^99 +2 divided by (9+1) , the remainder is 1
so 9^99 divided by 10 ,the remainder is 1-2 =-1 i.e the remainder is 10-1=9