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David Terr
Ph.D. Math, UC Berkeley

P.5. Polynomials

Polynomials are a very important class of functions in mathematics. In order to define a polynomial, we must start with some other definitions. A monomial is a product of a real number, known as a coefficient, and a finite number of variables each raised to a positive integer power. Examples of monomials are 1, -4, x, 3x2, -5xy, 2xyz, -x2y2, and πyz3. The degree of a monomial is the sum of powers of all variables in the monomial. A polynomial is a finite sum of monomials. The degree of a polynomial is the maximum degree of its monomials.

The simplest type of polynomials are polynomials of one variable, more commonly known as univariate polynomials. A univariate polynomial is an expression of the form anxn + an-1xn-1 + ... + a2x2 + a1x + a0, where the the numbers a0, a1, a2, ..., an are real valued constants known as the coefficients of the polynomial and n is a whole number known as the degree of the polynomial. Examples of univariate polynomials are x, 3x + 4, -4x3, x2 + 6x - 5, -4x3, and x3 - 3x2 + 6x - 6. A multivariate polynomial is a polynomial of two or more variables. Some examples of multivariate polynomials are x + y, 3x + 5y - 6, 4xy, x2 + xy - y2, xyz, and x3 - 3y2 + 5xz - 8.

Polynomials of small degree have special names. The following table lists the names of polynomials of degree up to 9:

degree name
0
constant
1
linear
2
3
cubic
4
quartic
5
quintic
6
sextic
7
septic
8
octic
9
nonic

Table P.5.1: Names of Polynomials of Small Degree

Example 1: Determine the degrees and names of the following polynomials and identify them as univariate or multivariate.

• (a) x
• (b) 3x8 + 15x4 + 1
• (c) π + 3
• (d) π + x
• (e) 2x + 5y + 4
• (f) -8xyz
• (g) x2y2 + 3xyz3 - 2z4

Solution:

• (a) The polynomial x is univariate of degree 1 (linear) since x = x1.
• (b) The polynomial 3x8 + 15x4 + 1 is univariate of degree 8 (octic) since 8 is the largest power of x in any of its monomials.
• (c) The polynomial π + 3 is univariate of degree 0 (constant) since it does not contain any variables.
• (d) The polynomial π + x is univariate of degree 1 (linear) since 1 is the largest power of x of any of its monomials.
• (e) The polynomial 2x + 5y + 4 is multivariate of degree 1 (linear) since the monomials 2x and 5y each have maximal degree 1. (The remaining monomial, 4, has degree 0.)
• (f) The polynomial -8xyz is multivariate of degree 3 (cubic) since it is a monomial with the sum of powers of its variables equal to 3.
• (g) The polynomial x2y2 + 3xyz3 - 2z4 is multivariate of degree 5 (quintic), the monomial 3xyz3 having maximal degree 1 + 1 + 3 = 5.

A polynomial may be evaluated at particular values of its variables. We illustrate this with a couple of examples.

Example 2: Evaluate the polynomial 3x2 + 5x + 6 at x=2.

Solution: We substitue 2 for x and solve, obtaining 3*22 + 5*2 + 6 = 3*4 + 5*2 + 6 = 12 + 10 + 6 = 28.

Example 3: Evaluate the polynomial x2 + 4xy - 5y2 at x=3; y=5.

Solution: Substituting these values we obtain 32 + 4*3*5 - 5*52 = 9 + 60 - 125 = -56.