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12.1 Fill in the gaps in the proof of the Rational Roots Theorem, which states that for all polynomials f(x)=anxn+an−1
Posted: Thu Jul 14, 2022 3:55 pm
by answerhappygod
- 12 1 Fill In The Gaps In The Proof Of The Rational Roots Theorem Which States That For All Polynomials F X An Xn An 1 1 (107.54 KiB) Viewed 42 times
12.1 Fill in the gaps in the proof of the Rational Roots Theorem, which states that for all polynomials f(x)=anxn+an−1xn−1+…+a1x+a0 with integer coefficients and n≥1, if dc is a rational root of f(x) with gcd(c,d)=1, then c∣a0 and d∣an. Proof. 12.1A Since an∈Zf(d)=0f(dc)=0f(cd)=0 we have an(dc)n+an−1(dc)n−1+…+a1(dc)+a0=0 12.1B Multiplying on both sides by cn−1dcdn−1dnc we get ancn+an−1cn−1d+…+a1cdn−1+a0dn=0. It follows that a0dn=−c(ancn−1+an−1cn−2d+…+a2cdn−2+a1dn−1) and hence c∣a0dn. Since gcd(c,d)=1, by the repeated use of Coprimeness and Divisibility, we have gcd(c,dn)=1. Since c∣a0dn and gcd(c,dn)=1, 12.1C by Coprimeness and Divisibility , Bézout's Lemma, Coprimeness Characterization Theorem, Euclid's Lemma, we have c∣a0. Similarly, we can prove d∣an. 12.2 List all the possible rational roots of f(x)=15x4+17x3+15x2−4 (There should be more than 20 possibilities in your list. You do not need to check which if any are actually roots). Enter your answer as a semicolon-separated list. The order does not matter. For example, "1;−1;3;−3 " is equivalent to "-3;1;-1;3". If needed, you may express roots as " 1/3" and "-1/3".