# Degree (of an Expression)

"Degree" can mean several things in mathematics:

- In Geometry a degree (°) is a way of measuring angles,
- But here we look at what degree means in
**Algebra**.

In Algebra "Degree" is sometimes called "Order"

## Degree of a Polynomial (with one variable)

A polynomial looks like this:

example of a polynomial this one has 3 terms |

The **Degree** (for a polynomial with one variable, like *x*) is:

... the **largest exponent** of that variable.

### More Examples:

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

The Degree is 3 (largest exponent of x) |
||

The Degree is 5 (largest exponent of x) |
||

The Degree is 2 (largest exponent of z) |

## Names of Degrees

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

Degree | Name | Example |
---|---|---|

0 | Constant | 7 |

1 | Linear | x+3 |

2 | Quadratic | x^{2}−x+2 |

3 | Cubic | x^{3}−x^{2}+5 |

4 | Quartic | 6x^{4}−x^{3}+x−2 |

5 | Quintic | x^{5}−3x^{3}+x^{2}+8 |

Example: **y = 2x + 7** has a degree of 1, so it is a **linear** equation

Example: **5w ^{2} − 3** has a degree of 2, so it is

**quadratic**

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:

example of a polynomial with more than one variable |

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.

### Example: what is the degree of this polynomial:

Checking each term:

**5xy**has a degree of^{2}**3**(x has an exponent of 1, y has 2, and 1+2=3)**3x**has a degree of**1**(x has an exponent of 1)**5y**has a degree of^{3}**3**(y has an exponent of 3)**3**has a degree of 0 (no variable)

The largest degree of those is 3 (in fact two terms have a degree of 3), so the polynomial has a degree of **3**

### Example: what is the degree of this polynomial:

4z^{3} + 5y^{2}z^{2} + 2yz

Checking each term:

**4z**has a degree of^{3}**3**(z has an exponent of 3)**5y**has a degree of^{2}z^{2}**4**(y has an exponent of 2, z has 2, and 2+2=4)**2yz**has a degree of**2**(y has an exponent of 1, z has 1, and 1+1=2)

The largest degree of those is 4, so the polynomial has a degree of **4**

## Writing it Down

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

## When Expression is a 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 top (numerator) and subtracting the degree of the bottom (denominator).

Here are three examples:

## Calculating Other Types of Expressions

**Warning: Advanced Ideas Ahead!**

We can sometimes work out the degree of an expression by dividing ...

- the logarithm of the function by
- the logarithm of the variable

... then do that for larger and larger values, to see where the answer is "heading".

*(More correctly we should work out the Limit to Infinity of ln(f(x))/ln(x), but I just want to keep this simple here).*

Note: " |

Here is an example:

### Example: What is the degree of (3 plus the square root of x) ?

Let us try increasing values of x:

x | ln() | ln(x) | ln() / ln(x) |
---|---|---|---|

2 | 1.48483 | 0.69315 | 2.1422 |

4 | 1.60944 | 1.38629 | 1.1610 |

10 | 1.81845 | 2.30259 | 0.7897 |

100 | 2.56495 | 4.60517 | 0.5570 |

1,000 | 3.54451 | 6.90776 | 0.5131 |

10,000 | 4.63473 | 9.21034 | 0.5032 |

100,000 | 5.76590 | 11.51293 | 0.5008 |

1,000,000 | 6.91075 | 13.81551 | 0.5002 |

Looking at the table:

- as
**x**gets larger then**ln() / ln(x)**gets closer and closer to**0.5**

So the Degree is 0.5 (in other words 1/2)

*(Note: this agrees nicely with x ^{½} = square root of x, see Fractional Exponents)*

## Some Degree Values

Expression | Degree |
---|---|

log(x) | 0 |

e^{x} |
∞ |

1/x | −1 |

1/2 |