Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0 - it has no variable.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

Let's see how this works by looking at several polynomials. We'll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Monomial
Degree
\(14\)
\(0\)
 \(8 y^{2}\)
\(2\)		 \(-9 x^{3} y^{5}\)
\(8\)		 \(-13a\)
\(1\)
Binomial
Degree of each term
Degree of polynomial
\(a+7\)
\(1 \qquad 0\)
\(1\)
 \(4 b^{2}-5 b\)
\(2 \qquad 1\)
\(2\)		 \(x^{2} y^{2}-16\)
\(4 \qquad 0\)
\(4\)		 \(3 n^{3}-9 n^{2}\)
\(3 \qquad 2\)
\(3\)
Trinomial
Degree of each term
Degree of polynomial		 \(x^{2}-7 x+12\)
\(2 \qquad 1 \qquad 0\)
\(2\)		 \(9 a^{2}+6 a b+b^{2}\)
\(2 \qquad 2 \qquad 2\)
\(2\)		 \(6 m^{4}-m^{3} n^{2}+8 m n^{5}\)
\(4 \qquad 5 \qquad 6\)
\(6\)		 \(z^{4}+3 z^{2}-1\)
\(\begin{array}{lll}4 & 2 & 0\end{array}\)
\(4\)
Polynomial
Degree of each term
Degree of polynomial		 \(b+1\)
\(1 \qquad 1 \qquad 0\)
\(1\)		 \(4 y^{2}-7 y+2\)
\(2 \qquad 1 \qquad 1 \qquad 0\)
\(2\)		 \(4 x^{4}+x^{3}+8 x^{2}-9 x+1\)
\(\begin{array}{lllll}4 & 3 & 2 & 1 & 0\end{array}\)
\(4\)

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Example 6.2

Find the degree of the following polynomials.

  1. \(10 y\)
  2. \(4 x^{3}-7 x+5\)
  3. \(-15\)
  4. \(-8 b^{2}+9 b-2\)
  5. \(8 x y^{2}+2 y\)
Solution
  1. The exponent of \(y\) is one. \(y=y^{1}\)
\(10y\)
The degree is 1.
  1. The highest degree of all the terms is 3.
\(4 x^{3}-7 x+5\)
The degree is 3.
  1. The degree of a constant is 0.
\(−15\)
The degree is 0.

  1. The highest degree of all the terms is 2.
 \(-8 b^{2}+9 b-2\)
The degree is 2.
  1. The highest degree of all the terms is 3.
\(8 x y^{2}+2 y\)
The degree is 3.
Try It 6.3

Find the degree of the following polynomials:

  1. \(-15 b\) 
  2. \(10 z^{4}+4 z^{2}-5\)
  3. \(12 c^{5} d^{4}+9 c^{3} d^{9}-7\)
  4. \(3 x^{2} y-4 x\)
  5. \(-9\)
Try It 6.4

Find the degree of the following polynomials:

  1. \(52\)
  2. \(a^{4} b-17 a^{4}\)
  3. \(5 x+6 y+2 z\)
  4. \(3 x^{2}-5 x+7\)
  5. \(-a^{3}\)
Back to '8.2: Adding and Subtracting Polynomials'