P-x-x-n-a-n-1-x-n-1-a-1-x-a-0-be-a-polynomial-with-all-the-real-roots-prove-that-n-1-a-n-1-2-2na-n-2-

Tinku Tara June 4, 2023 Algebra 0 Comments

Question Number 33834 by uddipan last updated on 25/Apr/18

P(x)=x^n +a_(n−1) x^(n−1) +.... a_1 x+a_0 be a polynomial with all the real roots, prove that (n−1)a_(n−1) ^2 ≥ 2na_(n−2) .

$${P}\left({x}\right)={x}^{{n}} +{a}_{{n}−\mathrm{1}} {x}^{{n}−\mathrm{1}} +….\:{a}_{\mathrm{1}} {x}+{a}_{\mathrm{0}} \:\:\:{be}\:{a}\:{polynomial}\:{with}\:{all}\:{the}\:{real}\:{roots}, \\ $$$${prove}\:{that}\:\:\:\:\:\:\:\:\:\left({n}−\mathrm{1}\right){a}_{{n}−\mathrm{1}} ^{\mathrm{2}} \:\geqslant\:\mathrm{2}{na}_{{n}−\mathrm{2}} \:\:. \\ $$

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P-x-x-n-a-n-1-x-n-1-a-1-x-a-0-be-a-polynomial-with-all-the-real-roots-prove-that-n-1-a-n-1-2-2na-n-2-

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