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 |
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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 |
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4 |
1 |
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1, 2, 3 |
3 |
3 |
One root of first derivative equals one root of second
derivative. |
Graph C |

3 |
2 |
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3 |
1 |
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1 |
2 |
3 |
Both roots of first derivative equal two roots of
second derivative. |
Graph D |

2 |
2 |
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2 |
1 |
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1 |
1 |
3 |
Twice repeated root of first derivative equals one
root of second derivative. |
Graph E |

1 |
2 |
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1 |
1 |
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0 |
3 |
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0 |
2 |
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0 |
1 |

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