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Let a, b, c and d be distinct real numbers. Show that the equation (x – b) (x – c)(x - d) + (x – a)(x – c)(x – d) + (x – a)(x – b)(x - d) + (x - a)(x – b)(x – c) = 0 (1) has exactly 3 distinct real solutions. (Hint: Let p(x) = (x – a)(x – b)(x – c)(x – d). Then p(x) = 0 has how many distinct real solutions? Then use logarithmic differentiation to show that p' (x) is given by the expression on the left hand side of (1). Now, apply Rolle's theorem. )

Let a, b, c and d be distinct real numbers. Show that the equation (x – b) (x – c)(x - d) + (x – a)(x – c)(x – d) + (x – a)(x – b)(x - d) + (x - a)(x – b)(x – c) = 0 (1) has exactly 3 distinct real solutions. (Hint: Let p(x) = (x – a)(x – b)(x – c)(x – d). Then p(x) = 0 has how many distinct real solutions? Then use logarithmic differentiation to show that p' (x) is given by the expression on the left hand side of (1). Now, apply Rolle's theorem. )

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 21RE
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Please show apply rolle theorem 

Let a, b, c and d be distinct real numbers. Show that the equation (x – b)(x – c)(x – d) + (x – a)(x – c)(x – d) + (x – a)(x – b)(x – d) + (x – a)(x – b)(x – c) = 0 (1) has exactly 3 distinct real solutions. (Hint: Let p(x) = (x – a)(x – b) (x – c)(x – d). Then p(x) = 0 has how many distinct real solutions? Then use logarithmic differentiation to show that p' (x) is given by the expression on the left hand side of (1). Now, apply Rolle's theorem. )
Transcribed Image Text:Let a, b, c and d be distinct real numbers. Show that the equation (x – b)(x – c)(x – d) + (x – a)(x – c)(x – d) + (x – a)(x – b)(x – d) + (x – a)(x – b)(x – c) = 0 (1) has exactly 3 distinct real solutions. (Hint: Let p(x) = (x – a)(x – b) (x – c)(x – d). Then p(x) = 0 has how many distinct real solutions? Then use logarithmic differentiation to show that p' (x) is given by the expression on the left hand side of (1). Now, apply Rolle's theorem. )
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