Question

Show that if r is a positive odd integer then the polynomial \[x ^{r}+1\] is dividable by \[x+1\]

Answer 1

Let f(x)=x^r +1
Now, use remainder theorem

Answer 2

Can u?

Answer 3

Don't know the remainder theorem :/

Answer 4

If a polynomial f(x) is divided by (x-a) then (x-a) is a factor of f(x) if f(a)=0.

Answer 5

Oh sorry... it is Factor Theorem.... Now use this factor theorem.

Answer 6

I do get the factor theorem, so that proves a part of it. I do have the answer though and it says that this it only valid if and only if (-1)^r+1=0, what's the meaning of that?

Answer 7

f(x)=x^r +1
Now, when f(x) id divided by (x+1) then,
Remainder = f(-1)=(-1)^r+1
Since r is odd (-1)^r=-1
So,
Remainder =f(-1)=-1+1=0
Thus,x^r +1 is divisible by x+1

Answer 8

Oh, how stupid, I see that it's just a continuation of the factor theorm, thank you :)

Answer 9

Welcome