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

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sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.1Let f(x)=x^r +1 Now, use remainder theorem

frx
 3 years ago
Best ResponseYou've already chosen the best response.0Don't know the remainder theorem :/

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.1If a polynomial f(x) is divided by (xa) then (xa) is a factor of f(x) if f(a)=0.

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.1Oh sorry... it is Factor Theorem.... Now use this factor theorem.

frx
 3 years ago
Best ResponseYou've already chosen the best response.0I 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?

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.1f(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

frx
 3 years ago
Best ResponseYou've already chosen the best response.0Oh, how stupid, I see that it's just a continuation of the factor theorm, thank you :)
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