# Degree of Polynomial

Degree can mean several things in mathematics:

• In Geometry a degree (°) is a way of measuring angles,
• In Algebra “Degree” is sometimes called “Order”

## Degree of a Polynomial

A polynomial looks like this:

2x3 + 3x2 + x -1

The Degree (for a polynomial with one variable, like x) is the largest exponent of that variable.

More Examples:

 2x The Degree is 1 (a variable without anexponent actually has an exponent of 1) 2x3 + 3x2 + x -1 The Degree is 3 (largest exponent of x) 3y2 + y5 − 1 The Degree is 5 (largest exponent of y) 3 – z5 + 4z7 The Degree is 7 (largest exponent of z)

## Names of Degrees

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

 Degree Name Example 0 Constant 4 1 Linear x – 5 2 Quadratic x2−2x+4 3 Cubic 2x3−x2+3 4 Quartic 3x4−2x3+x−4 5 Quintic x5−4x3+3x2+2x

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:

xy2−2x+4

For each term:

1. Find the degree by adding the exponents of each variable in it,
2. The largest such degree is the degree of the polynomial.
Example

xy2−2x+4

Checking each term:

• xy2 has a degree of 3 (x has an exponent of 1, y has 2, and 1+2=3)
• 2x has a degree of 1 (x has an exponent of 1)
• 4 has a degree of 0 (no variable)

The largest degree of those is 3 , so the polynomial has a degree of 3

## Degree of a Polynomial in 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 numerator and subtracting the degree of the denominator.

Here are 3 examples

 Fraction Degree of Numerator Degree of Denominator Subtracting the degree Degree of Fraction  3 2 3-2 1  2 2 2-2 0  3 4 3-4 -1

## How to Writing it

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

deg(2x3−x2+3) = 3

## Degree Values

 Expression Degree 1/x −1 log(x) 0 √x ½ ex ∞