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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:

4x 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:

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: 5y2 - 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