A **Polynomial** can be expressed in terms that only have positive integer exponents and the operations of addition, subtraction, and multiplication. In other words, it must be possible to write the expression without division. It's easiest to understand what makes something a polynomial equation by looking at examples and non examples as shown below.

**Examples** of Polynomials

Example Polynomial | Explanation |
---|---|

x^{2} + 2x +5 |
Since all of the variables have integer exponents that are positive this is a polynomial. |

5x +1 | Since all of the variables have integer exponents that are positive this is a polynomial. |

(x^{7} + 2x^{4} - 5) * 3x |
Since all of the variables have integer exponents that are positive this is a polynomial. |

5x^{-2} +1 |
Not a polynomial because a term has a negative exponent |

3x^{½} +2 |
Not a polynomial because a term has a fraction exponent |

(5x +1) ÷ (3x) | Not a polynomial because of the division |

(6x^{2} +3x) ÷ (3x) |
Is actually a polynomial because it's possible to simplify this to 3x + 1 --which of course satisfies the requirements of a polynomial. (Remember the definition states that the expression 'can' be expressed using addition,subtraction, multiplication. So, if it's possible to simplify an expression into a form that uses only those operations and whose exponents are all positive integers...then you do indeed have a polynomial equation) |

**Polynomial Equation**- is simply a polynomial that has been set equal to zero in an equation.

### Polynomials *vs* Polynomial Equations

See the next set of examples to understand the difference

Polynomial | Polynomial Equations |
---|---|

x^{2} + 2x +5 |
0= x^{2} + 2x +5 |

5x +1 | 0 = 5x +1 |