12. Generalize Theorem 1.3.9 by proving that every rational solution of a polyno-mial equation-1x"-1+ «1x + a• = 0,+...with integer coefficients a k,is an integer solution. Theorem 1.3.9. If k is an integer and the equation x = k has a rational solution,then that solution is actually an integer.Proof. Suppose r is a ratiomal number such that r2 = k. Let r =as a fraction in which n and m have no common factors. Then,er expressed2nkand so n? = m²k.mThis equationm implies that m divides n2. However, if m 7 1, then m can beexpressed as a product of primes, and each of these primes must also divide n2.However, if a prime number divides n2, it must also divide n (Exercise 1.3.14).Thus, each prime factor of m divides n. Since n and m have no common factors,this is impossible. We conclude that m =1 and, hence, that r = n is an integer.

Give polynomial is xn+an-1xn-1+…+a1x+a0=0, where all the coefficient's ak are integers. Let p,q are…

Answered: Generalize Theorem 1.3.9 by proving…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Generalize Theorem 1.3.9 by proving that every rational solution of a polyno- mial equation x" + an-1a"-1 + + a1x + = 0, | with integer coefficients ak, is an integer solution.