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LOGARITHMIQUES.
ou
![{\displaystyle x^{4}+(m+p)x^{3}+(n+mp+q)x^{2}+(np+mq)x+nq=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a12d7ed9cd84f61f814ae848073d1fc95a1572e)
;
et, pour faire disparaître le quatrième terme, supposons
![{\displaystyle np+mq=0{\text{ ou }}q=-{\frac {np}{m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149b332a5484f7bb945579fd9025a3f44d1f0e25)
;
la résultante sera alors divisible par
, et ses deux racines effectives seront
![{\displaystyle x=-{\frac {m+p}{2}}\pm {\sqrt {{\frac {(m-p)^{2}}{4}}-{\frac {n(m-p)}{m}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e7c299e76ad29889d9965343b897c6f6a77f557)
.
Si, pour rendre rationnelles ces expressions, on fait
![{\displaystyle {\frac {(m-p)^{2}}{4}}-{\frac {n(m-p)}{m}}=\left[{\frac {m-p}{2}}-{\frac {n\delta }{m}}\right]^{2}={\frac {(m-p)^{2}}{4}}-{\frac {2n\delta (m-p)}{m}}+{\frac {n^{2}\delta ^{2}}{m^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/119ccd53ac21f591678ab819a0736ec291c34b75)
il viendra
![{\displaystyle n={\frac {m(m-p)(\delta -1)}{\delta ^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43989e9396353c12894f023651f575318a45c2c7)
![{\displaystyle q=-{\frac {p(m-p)(\delta -1)}{\delta ^{2}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76a3e67ca5f11a097de9f7401188422906cc49b2)
et
![{\displaystyle x=-{\frac {m+p}{2}}\pm \left({\frac {m-p}{2}}-{\frac {n\delta }{m}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/181e78a5db05c439f0ac1cdbdbcc2dbc55e82ab1)
,
d’où
![{\displaystyle x=-p-{\frac {n\delta }{m}}=-{\frac {p+m(\delta -1)}{\delta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c09ff653b4dee4f15769994f45083f12b82bf4)
,
et
![{\displaystyle x=-m+{\frac {n\delta }{m}}=-{\frac {m+p(\delta -1)}{\delta }}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d091bfe65afe4c7c041545ebbdf1ba771f95305c)
on aura, par conséquent, pour la résultante
![{\displaystyle x^{2}\left[x+{\frac {p+m(\delta -1)}{\delta }}\right]\left[x+{\frac {m+p(\delta -1)}{\delta }}\right]=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b869d071062e154b0f95195d5df85a88e8feb85f)
et pour l’équation principale
![{\displaystyle \left[x^{2}+mx+{\frac {m(m-p)(\delta -1)}{\delta ^{2}}}\right]\left[x^{2}+px-{\frac {p(m-p)(\delta -1)}{\delta ^{2}}}\right]=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02859ea95c68e6a1237c4d3218cd4602218abcd3)