Explanation Formula Used: The solution of cubic equation of the form x 3 + a x 2 + b x + c = 0 is given by, A 3 + B 3 − a 3 , ω A 3 + ω 2 B 3 − a 3 , and ω 2 A 3 + ω B 3 − a 3 , Where, ω = − 1 + i 3 2 , p = b − a 2 3 , q = c − a b 3 + 2 a 3 27 A = − q 2 + ( q 2 ) 2 + ( p 3 ) 3 , B = − q 2 − ( q 2 ) 2 + ( p 3 ) 3 Select A and B such that A 3 B 3 = − p 3 . Calculation: The given equation is x 3 − 12 x − 20 = 0 . Compare the given equation with x 3 + a x 2 + b x + c = 0 . Here a = 0 , b = − 12 , and c = − 20 . The value of ω is given by, ω = − 1 + i 3 2 Substitute 0 for a and − 12 for b and solve for the value of p , p = b − a 2 3 p = − 12 − 0 2 3 p = − 12 Substitute 0 for a , − 12 for b and − 20 for c and solve for the value of q , q = c − a b 3 + 2 a 3 27 q = − 20 − 0 × ( − 12 ) 3 + 2 ( 0 ) 3 27 q = − 20 Substitute − 12 for p and − 20 for q and solve for the value of A , A = − q 2 + ( q 2 ) 2 + ( p 3 ) 3 = − ( − 20 ) 2 + ( − 20 2 ) 2 + ( − 12 3 ) 3 = 10 + ( − 10 ) 2 + ( − 4 ) 3 = 16 Substitute − 12 for p and − 20 for q and solve for the value of B , B = − q 2 − ( q 2 ) 2 + ( p 3 ) 3 = − ( − 20 ) 2 − ( − 20 2 ) 2 + ( − 12 3 ) 3 = 10

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.5, Problem 3E

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Chapter 8.5, Problem 2E

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Chapter 8 Solutions

Elements Of Modern Algebra

Ch. 8.1 - True or False Label each of the following...Ch. 8.1 - Prob. 2TFE Ch. 8.1 - Prob. 3TFE Ch. 8.1 - Prob. 4TFE Ch. 8.1 - Prob. 5TFE Ch. 8.1 - Prob. 6TFE Ch. 8.1 - Prob. 7TFE Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E

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The solutions of the equation x 3 − 12 x − 20 = 0 .

Solution Summary: The author calculates the solutions of the cubic equation x3-12x-20=0.

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Chapter 8 Solutions