Degree (of an Expression)
"Degree" is sometimes called "Order"
Degree of a Polynomial (One Variable)
The Degree of a Polynomial with one variable (like x) is the largest exponent of the variable.
Examples:

The Degree is 1 (a variable without an exponent 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) 
Degree of a Polynomial (More Than One Variable)
If there is more than one variable in the polynomial, you need to look at each term (terms are separated by + or  signs):
 Find the degree of each term 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:
 5xy^{2} has a degree of 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^{3} has a degree of 3 (y has an exponent of 3)
 3 has a degree of 0 (no variable)
The largest is 3, so the polynomial has a degree of 3
Names of Degrees
When you know the degree you can also give it a name!
0 
constant 
1 
linear 
2 
quadratic 
3 
cubic 
4 
quartic 
5 
quintic 
Example: 5y^{2}  3 has a degree of 2, so it is quadratic
When Expression is a Fraction
You 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!
You can sometimes work out the degree of an expression by dividing ...

the logarithm of the function by

the logarithm of the variable
... for larger and larger values, to see where the answer is "heading".
(More correctly you should evaluate the Limit to Infinity of log(f(x))/log(x), but I just want to keep this simple here).
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 
log() 
log(x) 
log()
/log(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 log() / log(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 
