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​

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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 An Xn An 1 1 (107.54 KiB) Viewed 41 times

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​xn−1+…+a1​x+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 an​cn+an−1​cn−1d+…+a1​cdn−1+a0​dn=0. It follows that a0​dn=−c(an​cn−1+an−1​cn−2d+…+a2​cdn−2+a1​dn−1) and hence c∣a0​dn. Since gcd(c,d)=1, by the repeated use of Coprimeness and Divisibility, we have gcd(c,dn)=1. Since c∣a0​dn 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".