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:
- 5xy2 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)
- 5y3 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: 5xy2 - 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 x½ = square root of x, see Fractional Exponents)
Some Degree Values
| Expression |
Degree |
| log(x) |
0 |
| ex |
∞ |
| 1/x |
-1 |
 |
1/2 |
|
Home
