Degree (of an Expression)

"Degree" can mean several things in mathematics:

In Algebra "Degree" is sometimes called "Order"

Degree of a Polynomial (with one variable)

A polynomial looks like this:

polynomial example
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.

polynomial

More Examples:

4x   The Degree is 1 (a variable without an
exponent actually has an exponent of 1)
     
4x3 − x + 3   The Degree is 3 (largest exponent of x)
     
x2 + 2x5 − x   The Degree is 5 (largest exponent of x)
     
z2 − z + 3   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 x2−x+2
3 Cubic x3−x2+5
4 Quartic 6x4−x3+x−2
5 Quintic x5−3x3+x2+8

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

Example: 5w2 − 3 has a degree of 2, so it is quadratic

Higher order equations are usually harder to solve:

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:

polynomial example
example of a polynomial
with more than one variable

For each term:

The largest such degree is the degree of the polynomial.

Example: what is the degree of this polynomial:

polynomial

Checking each term:

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:

4z3 + 5y2z2 + 2yz

Checking each term:

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:

degree notation

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 ...

... 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: "ln" is the natural logarithm function.

  calculator ln button

Here is an example:

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

Let us try increasing values of x:

x ln(3+sqrt(x)) ln(x) ln(3+sqrt(x))
/ 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:

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
sqrt(x) 1/2