Polynomials

Introduction

A polynomial has one or more terms, where each term can be a constant, a variable, or a product of constants and/or variables with only positive integer exponents.

Examples:

$-8$

$-4x^3y$

$4x-3x^2$

$3x^2y+4x-7xy$

A polynomial of one term is called a monomial. In the above examples, the first two expressions are monomials.

A polynomial of two terms is called a binomial. The third expression in the above examples is a binomial.

A polynomial of three terms is called a trinomial. The last expression in the examples is a trinomial.

The sum of the exponents of the variables in a term is called the degree of the term. The highest degree among the terms of a polynomial is called the degree of the polynomial.

In the examples above, the first one, $-8$, has degree $0$, since there is no variable.

The degree of the second polynomial, $-4x^3y$, is $4$, since the sum of the exponents of the variables is $3+1=4$.

The polynomial $4x-3x^2$ has two terms: the degree of the first term is $1$ and that of the second is $2$. The highest is $2$, so $2$ is the degree of the polynomial.

The polynomial $3x^2y+4x-7xy$ has three terms: the degree of the first term is $3$, and those of the second and third terms are $1$ and $2$, respectively. So the highest degree is $3$, which is the degree of the polynomial.

Sections in this chapter

Exponents and scientific notation

Integer exponents and their properties — product, quotient, power, zero, and negative exponents — and writing numbers in scientific notation.

Operations on polynomials

Adding, subtracting, multiplying, and dividing polynomials, including polynomial long division.

Factoring polynomials

Greatest common factor, factoring by grouping, trinomials, perfect-square trinomials, difference of squares, and sum and difference of cubes.

Solving polynomial equations

Solving polynomial and quadratic equations by factoring using the zero product rule.