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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272501.png" /></td> </tr></table>
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$$ax^3+bx^2+cx+d = 0$$
 
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where $a\ne 0$. Replacing $x$ in this equation by the new unknown $y$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272502.png" />. Replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272503.png" /> in this equation by the new unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272504.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272505.png" />, one brings the equation to the following simpler (canonical) form:
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defined by $x=y-b/3a$, one brings the equation to the following simpler
 
+
(canonical) form:  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272506.png" /></td> </tr></table>
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$$y^3+py+q = 0$$
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where
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$$p=-\frac{b^2}{3a^2} + \frac{c}{a},$$
  
where
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$$q=\frac{2b^3}{27a^3}-\frac{bc}{3a^2}+\frac{d}{a}$$
 +
and the solution to this equation may be obtained by using
 +
Cardano's formula (cf.
 +
[[Cardano formula|Cardano formula]]); in other words, any cubic
 +
equation is solvable in radicals.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272507.png" /></td> </tr></table>
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In Europe, the cubic equation was first solved in the 16th century. At the
 
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beginning of that century, S. Ferro solved the equation $x^3+px=q$, where
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272508.png" /></td> </tr></table>
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$p>0$, $q>0$, but did not publish his solution. N. Tartaglia rediscovered
 
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Ferro's result; he also solved the equation $x^3+px+q$ ($(p>0$, $q>0$), and
and the solution to this equation may be obtained by using Cardano's formula (cf. [[Cardano formula|Cardano formula]]); in other words, any cubic equation is solvable in radicals.
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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
The cubic equation was first solved in the 16th century. At the beginning of that century, S. Ferro solved the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c0272509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725011.png" />, but did not publish his solution. N. Tartaglia rediscovered Ferro's result; he also solved the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725014.png" />), and announced without proof that the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027250/c02725017.png" />) could be reduced to that form. Tartaglia communicated his results to G. Cardano, who published the solution of the general cubic equation in 1545.
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G. Cardano, who published the solution of the general cubic equation
 +
in 1545.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh,   "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD>
 +
<TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD>
 +
</TR></table>
  
  
  
 
====Comments====
 
====Comments====
The history is treated in [[#References|[a2]]], where Cardano's name (wrongly) occurs as Cardan (Chapt. 12).
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The European history is treated in
 +
[[#References|[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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.W. Rouse Ball,   "A short account of the history of mathematics" , Dover, reprint  (1960) pp. 123–125</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD>
 +
<TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD>
 +
</TR><TR><TD valign="top">[a2]</TD>
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<TD valign="top">Howard Eves, "An introduction to the history of
 +
mathematics", Saunders College Publishing, Philadelphia, 6. ed. (1964)</TD>
 +
</TR></table>
 +
 
 +
[[Category:Field theory and polynomials]]
 +
[[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=15586
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article