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:

2xThe Degree is 1 (a variable without anexponent actually has an exponent of 1)
2x3 + 3x2 + x -1The Degree is 3 (largest exponent of x)
3y2 + y5 − 1The Degree is 5 (largest exponent of y)
3 – z5 + 4z7The Degree is 7 (largest exponent of z)

Names of Degrees

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

DegreeNameExample
0Constant4
1Linearx – 5
2Quadraticx2−2x+4
3Cubic2x3−x2+3
4Quartic3x4−2x3+x−4
5Quinticx5−4x3+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

FractionDegree of NumeratorDegree of DenominatorSubtracting the degreeDegree of Fraction
Degree of Fraction 1323-21
Degree of Fraction 2222-20
Degree of Fraction 3343-4-1

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

ExpressionDegree
1/x−1
log(x)0
√x½
ex

Learn More

Exponents

Negative Exponents

Fractional Exponents

Quadratic Equation Solver

Algebra Index

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