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Use calculus to show that x + 6x that f(x) = x³ + 6x Briefly, why does part (a) show that a = 2? Note that 10 + V108 = 10 + 6/3. Suppose that we know that the cube-root of 10 + V108 has the form c+ d/3 where c, d are integers. What are the values of c and d? (Hint: Compute the cube of c+ dV3.) Use a method similar to part (c) to compute the cube-root of -10 + V108. Compute a by subtracting the result of part (d) from the result of part (c). What do you obtain? = 20 has a unique real solution. (Hint: Show - 20 has a real root and that it must be unique.)

Use calculus to show that x + 6x that f(x) = x³ + 6x Briefly, why does part (a) show that a = 2? Note that 10 + V108 = 10 + 6/3. Suppose that we know that the cube-root of 10 + V108 has the form c+ d/3 where c, d are integers. What are the values of c and d? (Hint: Compute the cube of c+ dV3.) Use a method similar to part (c) to compute the cube-root of -10 + V108. Compute a by subtracting the result of part (d) from the result of part (c). What do you obtain? = 20 has a unique real solution. (Hint: Show - 20 has a real root and that it must be unique.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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10 + V108 – V 3 3 1. Cardano's cubic formula gives a x³ + 6x = 20. It is quickly checked that 2 is also a solution. A calculator suggests that a = 2. In this problem we give two separate verifications that a = 2. (a) Use calculus to show that x3 + 6x = 20 has a unique real solution. (Hint: Show that f(x) = x³ + 6x – 20 has a real root and that it must be unique.) (b) Briefly, why does part (a) show that a = 2? (c) Note that 10 + V108 = 10 + 6/3. Suppose that we know that the cube-root of 10 + V108 has the form c+ dV3 where c, d are integers. What are the values of c and d? (Hint: Compute the cube of c+ dv3.) (d) Use a method similar to part (c) to compute the cube-root of -10 + V108. (e) Compute a by subtracting the result of part (d) from the result of part (c). What do you obtain? –10+ V108 as a solution to - 2. In the 19-th century it was finally shown that polynomial equations of degree 5 and higher need not be "solvable by radicals." (a) Give an example of a 5-th degree polynomial equation that is solvable by radicals and give its real solution(s). (b) The equation x – 6x + 3 = 0 is not solvable by radicals. Sketch the important features of the graph of f(x) = x5 – real roots. - 6x +3 and give approximate values of its
Transcribed Image Text:10 + V108 – V 3 3 1. Cardano's cubic formula gives a x³ + 6x = 20. It is quickly checked that 2 is also a solution. A calculator suggests that a = 2. In this problem we give two separate verifications that a = 2. (a) Use calculus to show that x3 + 6x = 20 has a unique real solution. (Hint: Show that f(x) = x³ + 6x – 20 has a real root and that it must be unique.) (b) Briefly, why does part (a) show that a = 2? (c) Note that 10 + V108 = 10 + 6/3. Suppose that we know that the cube-root of 10 + V108 has the form c+ dV3 where c, d are integers. What are the values of c and d? (Hint: Compute the cube of c+ dv3.) (d) Use a method similar to part (c) to compute the cube-root of -10 + V108. (e) Compute a by subtracting the result of part (d) from the result of part (c). What do you obtain? –10+ V108 as a solution to - 2. In the 19-th century it was finally shown that polynomial equations of degree 5 and higher need not be "solvable by radicals." (a) Give an example of a 5-th degree polynomial equation that is solvable by radicals and give its real solution(s). (b) The equation x – 6x + 3 = 0 is not solvable by radicals. Sketch the important features of the graph of f(x) = x5 – real roots. - 6x +3 and give approximate values of its
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