9. High school algebra and Cauchy-Schwarz:(a) Recall that the quadratic polynomial ƒ(t) = at² + bt+c, with a, b, c =R, is ≥ 0 for all t or ≤ 0 for all t if and only if it has either noreal roots or one (double) real root. From the quadratic formula, thishappens if and only if b² - 4ac ≤ 0.(b) Define f(t) = (X + tY, X + tY), where t is a real variable. Explainwhy f(t) ≥ 0 for all t.(c) Using the algebraic properties of the inner product, write f(t) as aquadratic polynomial in t with coefficients a = (Y, Y), b = 2(X,Y),= (X, X).and c =(d) Using the last sentence of part (a), prove |(X, Y)| ≤ ||X|| ||Y||.

(a) Consider the quadratic polynomial ft=at2+bt+c with a,b,c∈ℝ. Without loss of generality a>0.…

Recall that the quadratic polynomial f(t) = at² +bt+c, with a, b, c € R, is ≥ 0 for all t or ≤ 0 for all t if and only if it has either no real roots or one (double) real root. From the quadratic formula, this happens if and only if b2 - 4ac ≤ 0. Define f(t) = (X + tY, X + tY), where t is a real variable. Explain why f(t) ≥ 0 for all t. Using the algebraic properties of the inner product, write f(t) as a quadratic polynomial in with coefficients a = (Y, Y), b = 2(X, Y), and c = = (X, X). Using the last sentence of part (a), prove |(X, Y)| ≤ ||X|| ||Y||.

Recall that the quadratic polynomial f(t) = at² +bt+c, with a, b, c € R, is ≥ 0 for all t or ≤ 0 for all t if and only if it has either no real roots or one (double) real root. From the quadratic formula, this happens if and only if b2 - 4ac ≤ 0. Define f(t) = (X + tY, X + tY), where t is a real variable. Explain why f(t) ≥ 0 for all t. Using the algebraic properties of the inner product, write f(t) as a quadratic polynomial in with coefficients a = (Y, Y), b = 2(X, Y), and c = = (X, X). Using the last sentence of part (a), prove |(X, Y)| ≤ ||X|| ||Y||.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 4E

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