Fifth Degree Polynomials

(Incomplete . . . )

Fifth degree polynomials are also known as quintic polynomials. Quintics have these characteristics:

Number of Real Roots
   

Notes

Click for example

1, 2, 3, 4, 5

4

3
 Roots of first and second derivatives  are all different. No symmetry.

Graph A

1, 2, 3, 4, 5

4

3
 Roots of first and second derivatives  are all different. Point symmetry.

Graph B
 

4

2
   
 

4

1
   

1, 2, 3

3

3
 One root of first derivative equals  one root of second derivative.

Graph C
 

3

2
   
 

3

1
   

1

2

3
 Both roots of first derivative equal  two roots of second derivative.

Graph D
 

2

2
   
 

2

1
   

1

1

3
 Twice repeated root of first derivative  equals one root of second derivative.

Graph E
 

1

2
   
 

1

1
   
 

0

3
   
 

0

2
   
 

0

1
   

Click on any of the images below for specific examples of the fundamental quintic shapes.