Apating tumbasan

Ang **apating tumbasan** () sa iisang aligin ay isang tumbasang maisusulat bilang damikay na ikaapat na antas (o digri) na tumbas sa 0. Samakatuwid baga, maisusulat ito sa anyong
ax^4+bx^3+cx^2+dx + e=0
kung saan a \neq 0.

Sa lahatan, ang apating tumbasan sa n aligin ay tumbasang maisusulat sa anyong
P(\underbrace{x, y , z ... a}_{n \text{ beses} }) = 0
Kung saan isang damikay ang *P* sa ikaapat na antas.
## Apating sanyo 
Kung ax^4+bx^3+cx^2+dx + e=0, makukuha ang mga kalutasan nito gamit ang mga tumbasan o sanyo sa ibaba:

:{x_{1}=-\frac{b}{4a}+\frac{1}{2}\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}-\frac{1}{2}\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}+\frac{-b^3+4abc-8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}}}}

:{x_{2}=-\frac{b}{4a}+\frac{1}{2}\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}+\frac{1}{2}\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}-\frac{b^3-4abc+8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}}}}

:{x_{3}=-\frac{b}{4a}-\frac{1}{2}\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}-\frac{1}{2}\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}+\frac{b^3-4abc+8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}}}}

:{x_{4}=-\frac{b}{4a}-\frac{1}{2}\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}+\frac{1}{2}\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}+\frac{b^3-4abc+8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]{2}(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]{2}a}}}}}
Katumbas naman ang talangi sa

:{\Delta=256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e-27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de+18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2}

## Mga sanggunian 
Kategorya:Mga polynomial