## Degree of Polynomial

* Degree* can mean several things in mathematics:

- In Geometry a degree (°) is a way of measuring angles,
- In Algebra “Degree” is sometimes called “Order”

Table of Contents

## Degree of a Polynomial

A polynomial looks like this:

2x^{3} + 3x^{2} + x -1

The Degree (for a polynomial with one variable, like x) is the **largest exponent** of that variable.

**More Examples:**

2x | The Degree is 1 (a variable without anexponent actually has an exponent of 1) |

2x^{3} + 3x^{2} + x -1 | The Degree is 3 (largest exponent of x) |

3y^{2} + y^{5} − 1 | The Degree is 5 (largest exponent of y) |

3 – z^{5} + 4z^{7} | The Degree is 7 (largest exponent of z) |

## Names of Degrees

When we know the degree we can also give it a name!

Degree | Name | Example |

0 | Constant | 4 |

1 | Linear | x – 5 |

2 | Quadratic | x^{2}−2x+4 |

3 | Cubic | 2x^{3}−x^{2}+3 |

4 | Quartic | 3x^{4}−2x^{3}+x−4 |

5 | Quintic | x^{5}−4x^{3}+3x^{2}+2x |

Higher order equations are usually harder to solve:

- Linear equations are easy to solve
- Quadratic equations are a little harder to solve
- Cubic equations are harder again, but there are formulas to help
- Quartic equations can also be solved, but the formulas are very complicated
- Quintic equations have no formulas, and can sometimes be unsolvable!

## Degree of a Polynomial with More Than One Variable

When a polynomial has more than one variable, we need to look at each term. Terms are separated by + or – signs:

xy^{2}−2x+4

For each term:

- Find the degree by adding the exponents of each variable in it,
- The largest such degree is the degree of the polynomial.

xy^{2}−2x+4

Checking each term:

- xy2 has a degree of 3 (x has an exponent of 1, y has 2, and 1+2=3)
- 2x has a degree of 1 (x has an exponent of 1)
- 4 has a degree of 0 (no variable)

The largest degree of those is 3 , so the polynomial has a degree of 3

## Degree of a Polynomial in Fraction

We can work out the degree of a rational expression (one that is in the form of a fraction) by taking the degree of the numerator and subtracting the degree of the denominator.

Here are 3 examples

Fraction | Degree of Numerator | Degree of Denominator | Subtracting the degree | Degree of Fraction |

3 | 2 | 3-2 | 1 | |

2 | 2 | 2-2 | 0 | |

3 | 4 | 3-4 | -1 |

## How to Writing it

Instead of saying “the degree of (whatever) is 3” we write it like this:

deg(2x^{3}−x^{2}+3) = 3

## Degree Values

Expression | Degree |

1/x | −1 |

log(x) | 0 |

√x | ½ |

e^{x} | ∞ |