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Difference between revisions of "Cubic equation"

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An algebraic equation of degree three, i.e. an equation of the form
 
An algebraic equation of degree three, i.e. an equation of the form
  
 
$$ax^3+bx^2+cx+d = 0$$
 
$$ax^3+bx^2+cx+d = 0$$
 
where $a\ne 0$. Replacing $x$ in this equation by the new unknown $y$
 
where $a\ne 0$. Replacing $x$ in this equation by the new unknown $y$
defined by $x=y-b/2a$, one brings the equation to the following simpler
+
defined by $x=y-b/3a$, one brings the equation to the following simpler
 
(canonical) form:  
 
(canonical) form:  
 
$$y^3+py+q = 0$$
 
$$y^3+py+q = 0$$
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====Comments====
 
====Comments====
The Eurpean history is treated in
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The European history is treated in
 
[[#References|[a2]]], chap. 8.  
 
[[#References|[a2]]], chap. 8.  
 
In this book also results concerning cubic equations from
 
In this book also results concerning cubic equations from
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A.D.), and the most remarkable treatment of the cubic by the Persian
 
A.D.), and the most remarkable treatment of the cubic by the Persian
 
mathematician Omar Khayyam (1024 -- 1123) are discussed.
 
mathematician Omar Khayyam (1024 -- 1123) are discussed.
 
 
 
  
 
====References====
 
====References====
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mathematics", Saunders College Publishing, Philadelphia, 6. ed. (1964)</TD>
 
mathematics", Saunders College Publishing, Philadelphia, 6. ed. (1964)</TD>
 
</TR></table>
 
</TR></table>
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[[Category:Field theory and polynomials]]
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[[Category:History of mathematics]]

Latest revision as of 17:32, 10 April 2018


An algebraic equation of degree three, i.e. an equation of the form

$$ax^3+bx^2+cx+d = 0$$ where $a\ne 0$. Replacing $x$ in this equation by the new unknown $y$ defined by $x=y-b/3a$, one brings the equation to the following simpler (canonical) form: $$y^3+py+q = 0$$ where $$p=-\frac{b^2}{3a^2} + \frac{c}{a},$$

$$q=\frac{2b^3}{27a^3}-\frac{bc}{3a^2}+\frac{d}{a}$$ and the solution to this equation may be obtained by using Cardano's formula (cf. Cardano formula); in other words, any cubic equation is solvable in radicals.

In Europe, the cubic equation was first solved in the 16th century. At the beginning of that century, S. Ferro solved the equation $x^3+px=q$, where $p>0$, $q>0$, but did not publish his solution. N. Tartaglia rediscovered Ferro's result; he also solved the equation $x^3+px+q$ ($(p>0$, $q>0$), and announced without proof that the equation $x^3+q=px$ ($p>0$, $q>0$) could be reduced to that form. Tartaglia communicated his results to G. Cardano, who published the solution of the general cubic equation in 1545.

References

[1] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)


Comments

The European history is treated in [a2], chap. 8. In this book also results concerning cubic equations from ancient Babylonia (2000 B.C.), ancient Chinese (Wang Hs'iao-t'ung, 625 A.D.), and the most remarkable treatment of the cubic by the Persian mathematician Omar Khayyam (1024 -- 1123) are discussed.

References

[a1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[a2] Howard Eves, "An introduction to the history of mathematics", Saunders College Publishing, Philadelphia, 6. ed. (1964)
How to Cite This Entry:
Cubic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cubic_equation&oldid=19686
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article