Question

In: Advanced Math

Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is...

Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is an integer.

Solutions

Expert Solution

solution:-

which means

Suppose is an integer and and so that the fraction is in reduced form

Therefore,

Substituting in the above expression, we have

That is, is a root of

If is not an integer, then

Thus, where is a prime number

Substituting in we get

Multiplying both sides by we get

Thus, we have

That is,

So that

Thus,

But this contradicts the fact that

Therefore, we must have where

Which is to say that the rational root of must be an integer

Colby Messinger answered 1 year ago

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