Loaʻa ka inoa o ka papa waena o ka pō mai kahi koi kula: pono ʻoe e haʻi i ke ʻano hoʻonā no nā hoʻohālikelike quadratic inā ʻoe e ala ana i ke aumoe. I ke kolu a me ka ha o na degere, ua oi loa aku ke koi o keia holoholona hihiu. Aia pū kekahi mau ʻōlelo paʻa no kēia me ka hoʻohana ʻana i ka helu helu kumu a me nā aʻa, akā ʻoi aku ka lōʻihi. Eia lākou i loko o nā coefficient kumu \(a,b,c,d,e\) , me ka ʻole o nā mea hoʻololi kōkua a me nā kūlana kūpono.
Hoʻopiha mau ka helu alakaʻi i \(a\ne0\) ; i ole ia e haule ke degere. Ke ʻimi nei mākou i nā zero paʻakikī a me nā mea he nui. Ma ka hihia laina \(n=1\) ua lawa nā coefficients \(a\) a me \(b\) : hiki ke hoʻoholo pololei i ka hoohalike no \(x\) a loaʻa iā ia hoʻokahi zero maʻalahi. Pili kēia i nā coefficient paʻakikī. No ka \(a=0\) akā naʻe, ʻaʻohe hopena no \(b\ne0\) ; he \(a=b=0\) , kēlā me kēia paʻakikī \(x\) he hoʻonā. No ka hoohalike laina maoli i loaa mai:
$$ax+b=0,\qquad a\ne0.$$
$$\boxed{x_k=-\frac{b}{a},\qquad k\in\{1\}.}$$
No ka \(n=2\) i loko o ka huinahāhāhā nā coefficients \(a,b,c\) . Me \(k=1,2\) \((-1)^k\) e hoʻololi i ka hōʻailona a e hoʻihoʻi i nā zeros ʻelua me nā mea he nui. No nā coefficients maoli ʻelua mau hoʻonā maoli ʻokoʻa no \(b^2-4ac>0\) , no ka \(b^2-4ac=0\) he pālua a no ka waiwai ʻole ʻelua hui hui pū: A laila hiki iā ʻoe ke koho i nā aʻa i \(\sqrt{b^2-4ac}=i\sqrt{4ac-b^2}\) \(i\) imaginary. \(i^2=-1\) ʻo ia. ʻAʻohe hihia hoʻokaʻawale no nā coefficient paʻakikī; pili ke kumu me na kumu huinahalike elua:
$$ax^2+bx+c=0,\qquad a\ne0.$$
$$\boxed{x_k=\frac{-b+(-1)^k\sqrt{b^2-4ac}}{2a},\qquad k\in\{1,2\}.}$$
No \(n=3\) ua kākau pololei ʻia ke ʻano o Cardano ma \(a,b,c,d\) . ʻO ka huaʻōlelo \(k\) he papa kuhikuhi hoʻonā wale nō: koho nā waiwai \(1,2,3\) i nā zeros ʻekolu, ʻaʻohe mea kōkua ʻē aʻe. Aia i loko o ka mana kahi kumu mua ʻekolu o ka lokahi . Hōʻike kēlā me kēia hōʻailona kumu i kekahi kumu paʻakikī; ʻO ka huaʻōlelo hou e hōʻike i ka waiwai like ma nā wahi a pau. Me \(b^2-3ac=0\) wale no ka waiwai o ke kumu huinahalike o loko \(2b^3-9abc+27a^2d\) , a i ole ia, lilo ka radicand o ke kumu huinahalike i ole.:
$$ax^3+bx^2+cx+d=0,\qquad a\ne0.$$
$$\left(b^2-3ac,\;2b^3-9abc+27a^2d\right)\ne(0,0).$$
$$\boxed{\begin{gathered}x_k=-\frac{b}{3a}-\frac{\left(\frac{-1+i\sqrt{3}}{2}\right)^{k-1}}{3a}\sqrt[3]{\frac{1}{2}\left(\begin{aligned}&2b^3-9abc+27a^2d\\&+\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}\end{aligned}\right)}-\frac{b^2-3ac}{3a\left(\frac{-1+i\sqrt{3}}{2}\right)^{k-1}\sqrt[3]{\frac{1}{2}\left(\begin{aligned}&2b^3-9abc+27a^2d\\&+\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}\end{aligned}\right)}}\\[5pt]k\in\{1,2,3\}.\end{gathered}}$$
\(x^3-1=0\) hōʻike i ke ʻano o ka hoʻoholo ʻana o ke kumu huinahā i loko: Eia \(b^2-3ac=0\) a me \(2b^3-9abc+27a^2d=-27\) . Me ka waiwai \(27\) ma kahi o \(-27\) ʻo ke kumu cube he \(\frac{-27+27}{2}=0\) a ʻaʻole hiki ke loiloi ʻia ke kumu. ʻO ke kūlana ma lalo o ka hoʻohālikelike mua e hoʻokaʻawale i ka hihia i helu ʻia nā huaʻōlelo ʻelua a ʻaʻole nalo ke kumu cube. A laila, loaʻa i ka hoʻohālikelike ʻekolu ʻekolu a ʻaʻole i uhi ʻia e ke kumu hoʻohālike i hāʻawi ʻia ma aneʻi.
No ka huina cubic maoli, o ka hoailona o \(b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd\) ka mea e hooholo ai i na zeros: O ka waiwai maikai, oia hoi ekolu mau hooluolu maoli like ole, o ke kumu waiwai ino, hookahi mea maoli a elua mea maoli ole, ma ka ole, he maoli no me ka hana hou ana. I ka hihia mua, hoʻoikaika kēia ʻano Cardano i nā waiwai waena paʻakikī. ʻAʻole i paʻa mau ka helu helu: aneane like nā waiwai waena hiki ke kūʻai i nā huahelu i nā helu lana; inā pono, hoʻohana ʻia ka pololei kiʻekiʻe.
No ka \(n=4\) ua kākau piha ʻia ke ʻano o Ferrari ma \(a,b,c,d,e\) . Holo ka papa kuhikuhi \(k\) mai \(1\) a i \(4\) ; na mana o \(-1\) e hopu i na hui hoailona eha a e hui pono i na aa. Hōʻike nā huaʻōlelo kumu like i ka waiwai like, ʻoiai no \(k\) ʻokoʻa. No ka \(c^2-3bd+12ae=0\) \(2c^3-9bcd+27b^2e+27ad^2-72ace\) ka waiwai o ke kumu huinahalike o loko. ʻO nā kūlana ma lalo o ka hoʻohālikelike mua e pale i ka mahele ʻana me ka ʻole; ua hoʻopaʻa ʻia kēlā me kēia zero me kona nui:
$$ax^4+bx^3+cx^2+dx+e=0,\qquad a\ne0.$$
$$\begin{gathered}b^3-4abc+8a^2d\ne0,\\\left(c^2-3bd+12ae,\;2c^3-9bcd+27b^2e+27ad^2-72ace\right)\ne(0,0).\end{gathered}$$
$$\boxed{\begin{gathered}x_k=-\frac{b}{4a}\\[6pt]+\frac{(-1)^{\frac{k(k-1)}{2}}\sqrt{3}}{12a}\sqrt{\begin{aligned}&3b^2-8ac+4a\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}\\&+\frac{4a\left(c^2-3bd+12ae\right)}{\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}}\end{aligned}}\\[6pt]+\frac{(-1)^k\sqrt{3}}{12a}\sqrt{\begin{aligned}&6b^2-16ac-4a\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}\\&-\frac{4a\left(c^2-3bd+12ae\right)}{\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}}\\&-\frac{6(-1)^{\frac{k(k-1)}{2}}\sqrt{3}\left(b^3-4abc+8a^2d\right)}{\sqrt{\begin{aligned}&3b^2-8ac+4a\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}\\&+\frac{4a\left(c^2-3bd+12ae\right)}{\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}}\end{aligned}}}\end{aligned}}\\[6pt]k\in\{1,2,3,4\}.\end{gathered}}$$
ʻO ke kūlana mua ke kāpae nei i nā polynomial e like me ke kūlana \(x=-\frac{b}{4a}\) , me nā hoohalike biquadratic me \(b=d=0\) e like me \(x^4-5x^2+4=0\) . Ma laila ke kumu nui mua hiki ke lilo i zero a laila i loko o ka denominator. Hoʻokuʻu ʻia ke kūlana ʻelua ma ka liʻiliʻi he ʻekolu ʻekolu, no ka laʻana me \(x^4-6x^2+8x-3=(x-1)^3(x+3)=0\) : Ma ʻaneʻi e nalowale ke kumu huipu ma ka denominator. ʻAʻole i uhi ʻia nā hihia kūikawā i kāpae ʻia e ka pahu pahu; ʻAʻole ʻae ʻia nā ʻōkuhi hoʻonā ʻē aʻe ma aneʻi.
Mai \(n=5\) ʻaʻole kōkua nā mākau hoʻopaʻanaʻau maikaʻi loa: ʻaʻohe kumu maʻamau i hana ʻia me kahi helu palena o nā hana helu a me nā kumu kumu. Penei ka olelo a ka Abel-Ruffini theorem ; hiki ke loaʻa kahi hōʻoia ma Milne's Fields and Galois Theory . Hiki ke hoʻoholo ʻia nā hoʻohālikelike pākahi me ka hoʻohana ʻana i nā radical. A e mau ana nā waiwai like ʻole a me nā hana kūikawā, e like me ka Bring's radical ma ka pae ʻelima. No laila ʻaʻole nalo nā zeros, ʻo ka manaʻo wale nō o kahi kumu kumu kumu honua.