Fifth Degree Polynomials

(Incomplete . . . )

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

• One to five roots.
• Zero to four extrema.
• One to three inflection points.
• No general symmetry.
• It takes six points or six pieces of information to describe a quintic function.
• Roots are not solvable by radicals (a fact established by Abel in 1820 and expanded upon by Galois in 1832).

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.