Explanation Procedure used: 1 . Find an equivalent inequality with P ( x ) on one side and 0 on the other. 2. Change the inequality symbol to an equal sign and solve the related equation; that is, solve P ( x ) = 0 . 3. Use the solutions to divide the x axis into intervals. Then select a test value from each interval and determine the sign of the polynomial on the interval. 4. Determine the intervals for which the inequality is satisfied and write intervals notation or set builder notation for the solution set. Include the endpoints of the intervals in the solution set if the inequality symbol is ≤ or ≥ . Calculation: The given polynomial inequality is x 4 − 6 x 2 + 5 > 0 . By the procedure from 1 to 4, find the solution set of the inequality. Change the inequality sign by equal sign to form equivalent equation as x 4 − 6 x 2 + 5 = 0 . In order to solve the equation, first factor it by grouping as follows. x 4 − 6 x 2 + 5 = 0 ( x 2 − 1 ) ( x 2 − 5 ) = 0 x 2 − 1 = 0 , x 2 − 5 = 0 x = ± 1 , x = ± 5 The solution set is { − 5 , − 1 , 1 , 5 } . The solutions divide the x axis into the intervals as ( − ∞ , − 5 ) , ( − 5 , − 1 ) , ( − 1 , 1 ) , ( 1 , 5 ) and ( 5 , ∞ ) . Find the sign of the polynomial on each interval by selecting test value as follows. Let the polynomial equation is denoted by f ( x ) = x 4 − 6 x 2 + 5 . Obtain the sign of polynomial f ( x ) = x 4 − 6 x 2 + 5 > at x = − 3 on the interval ( − ∞ , − 5 ) as follows. f ( − 3 ) = ( − 3 ) 4 − 6 ( − 3 ) 2 + 5 = 81 − 54 + 5 = 32 > 0 Obtain the sign of polynomial f ( x ) = x 4 − 6 x 2 + 5 > at x = − 2 on the interval ( − 5 , − 1 ) as follows

College Algebra: Graphs and Models (6th Edition)

6th Edition

ISBN: 9780134179032

Author: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Publisher: PEARSON

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Chapter 4.6, Problem 88E

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To solve: The polynomial inequality x4−6x2+5>0.

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Chapter 4.6, Problem 87E

Chapter 4.6, Problem 89E

Chapter 4 Solutions

College Algebra: Graphs and Models (6th Edition)

Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 10E

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