Polynomials

A polynomial is a particular type of algebraic expression that serves as a fundamental building block in algebra.

The expression 3x³ - 15x² + 7x - 2 is an example of a polynomial in one variable. Because this expression could be written as

3x³ + (-15x²) + 7x + (-2), we say that this polynomial is a sum of four terms: 3x³, -15x², 7x, and -2.

A term of a polynomial is a single number or the product of a number and one or more variables raised to whole number powers. The number preceding the variable in each term is called the coefficient of that variable. In 3x³ - 15x² + 7x - 2 the coefficient of x³ is 3, the coefficient of x² is -15, and the coefficient of x is 7. In algebra a number is frequently referred to as a constant, and so the last term -2 is called the constant term. A polynomial is defined as a single term or a sum of a finite number of terms.

 

Example 1

Identifying polynomials

Determine whether each algebraic expression is a polynomial.

a) -3

b) 3x +2^-1

c) 3x^-2 + 4y²

d)

e) x⁴⁹ - 8x² + 11x - 2

Solution

a) The number -3 is a polynomial of one term, a constant term.

b) Since 3x + 2^-1 can be written as , it is a polynomial of two terms.

c) The expression 3x^-2 + 4y²  is not a polynomial because x has a negative exponent.

d) If this expression is rewritten as x^-1 + x^-2, then it fails to be a polynomial because of the negative exponents. So a polynomial does not have variables in denominators, and is not a polynomial.

e) The expression x⁴⁹ - 8x² + 11x - 2  is a polynomial.

For simplicity we usually write polynomials in one variable with the exponents in decreasing order from left to right. Thus we would write 3x³ - 15x² + 7x - 2 rather than - 15x² - 2 + 7x + 3x³.

When a polynomial is written in decreasing order, the coefficient of the first term is called the leading coefficient.

Certain polynomials have special names depending on the number of terms. A monomial is a polynomial that has one term, a binomial is a polynomial that has two terms, and a trinomial is a polynomial that has three terms. The degree of a polynomial in one variable is the highest power of the variable in the polynomial. The number 0 is considered to be a monomial without degree because 0 = 0xⁿ, where n could be any number.

 

Example 2

Identifying coefficients and degree

State the degree of each polynomial and the coefficient of x². Determine whether the polynomial is monomial, binomial, or trinomial.

a)

b) x⁴⁸ - x²

c) 6

Solution

a) The degree of this trinomial is 3, and the coefficient of x² is .

b) The degree of this binomial is 48, and the coefficient of x² is -1.

c) Because 6 = 6x⁰, the number 6 is a monomial with degree 0. Because x² does not appear in this polynomial, the coefficient of x² is 0.

Although we are mainly concerned here with polynomials in one variable, we will also encounter polynomials in more than one variable, such as

4x² - 5xy + 6y², x² + y² + z², and ab² - c³.

In a term containing more than one variable, the coefficient of a variable consists of all other numbers and variables in the term. For example, the coefficient of x in -5xy is -5y, and the coefficient of y is -5x.