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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 a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀, where the the numbers a₀, a₁, a₂, ..., a_n 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	quadratic
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.

 

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