Editing Quintic function (section)

===Other solvable quintics===

There are infinitely many solvable quintics in Bring–Jerrard form which have been parameterized in a preceding section.

Up to the scaling of the variable, there are exactly five solvable quintics of the shape <math>x^5+ax^2+b</math>, which are<ref>{{cite web |first=Noam |last=Elkies |title=Trinomials {{nobr|a x{{sup|n}} + b x + c}} with interesting Galois groups |url=https://www.math.harvard.edu/~elkies/trinomial.html |publisher=[[Harvard University]]}}</ref> (where ''s'' is a scaling factor):
:<math>x^5-2s^3x^2-\frac{s^5}{5} </math>
:<math> x^5-100s^3x^2-1000s^5</math>
:<math>x^5-5s^3x^2-3s^5 </math>
:<math>x^5-5s^3x^2+15s^5 </math>
:<math> x^5-25s^3x^2-300s^5</math>

Paxton Young (1888) gave a number of examples of solvable quintics:
:{| <math>x^5+3x^2+2x-1 </math> ||
|-
| <math> x^5-10x^3-20x^2-1505x-7412</math> ||
|-
| <math>x^5+\frac{625}{4}x+3750 </math> ||
|-
| <math>x^5-\frac{22}{5}x^3-\frac{11}{25}x^2+\frac{462}{125}x+\frac{979}{3125} </math> ||
|-
| <math>x^5+20x^3+20x^2+30x+10 </math> || <math>~\qquad ~</math> Root: <math> \sqrt[5]{2}-\sqrt[5]{2}^2+\sqrt[5]{2}^3-\sqrt[5]{2}^4</math>
|-
|<math>x^5-20x^3+250x-400 </math> ||
|-
| <math>x^5-5x^3+\frac{85}{8}x-\frac{13}{2} </math> ||
|-
|<math> x^5+\frac{20}{17}x+\frac{21}{17}</math> ||
|-
|<math>x^5-\frac{4}{13}x+\frac{29}{65}</math> ||
|-
|<math> x^5+\frac{10}{13}x+\frac{3}{13}	</math> ||
|-
| <math> x^5+110(5x^3+60x^2+800x+8320)</math> ||
|-
| <math>x^5-20 x^3 -80x^2 -150x -656</math>||
|-
| <math> x^5 -40x^3 +160x^2 +1000x -5888</math> ||
|-
|<math> x^5-50x^3-600x^2-2000x-11200</math> ||
|-
| <math>x^5+110(5x^3 + 20x^2 -360x +800)</math> ||
|-
| <math> x^5 -20x^3 +170x + 208</math> ||
|}

An infinite sequence of solvable quintics may be constructed, whose roots are sums of {{mvar|n}}th [[roots of unity]], with {{nobr|{{math|''n'' {{=}} 10''k'' + 1}}}} being a [[prime number]]:

:{|
|-
| <math>x^5+x^4-4x^3-3x^2+3x+1</math> || || Roots: <math>2\cos\left(\frac{2k\pi}{11}\right)</math>
|-
| <math> x^5+x^4-12x^3-21x^2+x+5</math> || || Root: <math> \sum_{k=0}^5 e^\frac{2i\pi 6^k }{31}</math>
|-
| <math>x^5+x^4-16x^3+5x^2+21x-9</math> || || Root: <math>\sum_{k=0}^7 e^\frac{2i\pi 3^k }{41}</math>
|-
| <math>x^5+x^4-24x^3-17x^2+41x-13</math> || <math>~\qquad ~</math> || {{nowrap|1= Root: <math>\sum_{k=0}^{11} e^\frac{2i\pi (21)^k }{61}</math>}}
|-
| <math>x^5+x^4 - 28x^3 + 37x^2 + 25x + 1</math> || || {{nowrap|1= Root: <math>\sum_{k=0}^{13} e^\frac{2i\pi (23)^k }{71}</math>}}
|}

There are also two parameterized families of solvable quintics:
The Kondo–Brumer quintic,

:<math> x^5 + (a-3)\,x^4 + (-a+b+3)\,x^3 + (a^2-a-1-2b)\,x^2 + b\,x + a = 0 </math>

and the family depending on the parameters <math>a, \ell, m</math>
:<math>	x^5 - 5\,p \left( 2\,x^3 + a\,x^2 + b\,x \right) - p\,c = 0 </math>

where

::<math> p = \tfrac{1}{4} \left[\, \ell^2 (4m^2 + a^2) - m^2 \,\right] \;,</math>
:
::<math> b = \ell \, ( 4m^2 + a^2 ) - 5p - 2m^2 \;,</math>
:
::<math> c = \tfrac{1}{2} \left[\, b(a + 4m) - p(a - 4m) - a^2m  \,\right] \;.</math>