Suppose that x=a/bx=a/b is a rational root, wlog in lowest terms, i.e. a,ba,b are coprime. Then
0=bnT(a/b)=an+b(cn−1an−1+⋯+c0bn−1)0=bnT(a/b)=an+b(cn−1an−1+⋯+c0bn−1)
Thus b∣an.b∣an. If |b|>1|b|>1 then some prime p∣b∣an,p∣b∣an, so p∣a,p∣a, contra a,ba,b coprime. Therefore we conclude that |b|=1,|b|=1, so a/ba/bis an integer if rational.
Remark This is essentially one half of the well-known Rational Root Test. The other half follows by reciprocation symmetry, i.e. by applying this result to the reversed polynomial (satisfied by x−1).
10.28. Prove that a monic polynomial with integer coefficients can have periodic points with minimal period only 1, 2, or 4 10.28. Prove that a monic polynomial with integer coefficients ca...
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The Fibonacci Sequence F1, F2, ... of
integers is defined recursively by F1=F2=1
and Fn=Fn-1+Fn-2 for each integer
. Prove
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7.23)
n 3 Chapter 7 Reviewing Proof Techniques 196 an-2 for every integer and an ao, a1, a2,... is a sequence of rational numbers such that ao = n > 2, then for every positive integer n, an- 3F nif n is even 2Fn+1 an = 2 Fn+ 1 if n is odd....
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ONLY NUMBER 2
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Please prove the theorems,
thank you
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