1. From the Cardano reading and videos you can see that he can solve the "depressed cubic" (no x² term) in any of its variations. The general cubic can be turned into a depressed cubic, so he could solve all cubics. Verify this using modern notation: a. Start with the general cubic ax³ + bx² + cx+d = 0, make the substitution x = y-, b За and simplify. Gather all the like terms in y. Write each of your coefficients of the y terms (and constant term) as a single fraction (i.e., get a common denominator). Do not be afraid of a little algebra! Naturally, the 12 terms should cancel out. b. This should leave you with a "depressed cubic" which you should write in the form y³ + Cy=D. [Don't forget to divide by a.] Write out what C and D are equal to in terms of a, b, c, d. Do not be afraid of a little algebra!
1. From the Cardano reading and videos you can see that he can solve the "depressed cubic" (no x² term) in any of its variations. The general cubic can be turned into a depressed cubic, so he could solve all cubics. Verify this using modern notation: a. Start with the general cubic ax³ + bx² + cx+d = 0, make the substitution x = y-, b За and simplify. Gather all the like terms in y. Write each of your coefficients of the y terms (and constant term) as a single fraction (i.e., get a common denominator). Do not be afraid of a little algebra! Naturally, the 12 terms should cancel out. b. This should leave you with a "depressed cubic" which you should write in the form y³ + Cy=D. [Don't forget to divide by a.] Write out what C and D are equal to in terms of a, b, c, d. Do not be afraid of a little algebra!
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Italian Algebra
![1. From the Cardano reading and videos you can see that he can solve the "depressed cubic" (no
x² term) in any of its variations. The general cubic can be turned into a depressed cubic, so he
could solve all cubics.
Verify this using modern notation:
a. Start with the general cubic ax³ + bx² + cx+d = 0, make the substitution x = y-,
b
За
and simplify. Gather all the like terms in y. Write each of your coefficients of the y
terms (and constant term) as a single fraction (i.e., get a common denominator). Do
not be afraid of a little algebra! Naturally, the 12 terms should cancel out.
b. This should leave you with a "depressed cubic" which you should write in the form
y³ + Cy=D. [Don't forget to divide by a.] Write out what C and D are equal to in
terms of a, b, c, d. Do not be afraid of a little algebra!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda859fed-ae51-422c-9773-627805a70b82%2F592fd942-908e-4cc2-bb10-932ae8b017a5%2F5mwyvqv_processed.png&w=3840&q=75)
Transcribed Image Text:1. From the Cardano reading and videos you can see that he can solve the "depressed cubic" (no
x² term) in any of its variations. The general cubic can be turned into a depressed cubic, so he
could solve all cubics.
Verify this using modern notation:
a. Start with the general cubic ax³ + bx² + cx+d = 0, make the substitution x = y-,
b
За
and simplify. Gather all the like terms in y. Write each of your coefficients of the y
terms (and constant term) as a single fraction (i.e., get a common denominator). Do
not be afraid of a little algebra! Naturally, the 12 terms should cancel out.
b. This should leave you with a "depressed cubic" which you should write in the form
y³ + Cy=D. [Don't forget to divide by a.] Write out what C and D are equal to in
terms of a, b, c, d. Do not be afraid of a little algebra!
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