# Introduction to Algebraic Expression and Polynomials

An **algebraic Expressions **is an expression formed from any combination of numbers and variables by using the operations of addition, subtraction, multiplication, division, exponentiation (raising to powers), or extraction of roots.

For instance,,, , , , and are algebraic expressions. By an algebraic expression in certain variables, we mean an expression that contains only those variables, and by a **constant**, we mean an algebraic expression that contains no variables at all. If numbers are substituted for the variables in an algebraic expression, the resulting number is called the **value** of the expression for these values of the variables.

__Example__:

Find the value of when and .

__Solution__:

Substituting and , we obtain

If an algebraic expression consists of parts connected by plus or minus signs, it is called an **algebraic sum**, and each parts, together with the sign preceding it, is called a **term**. For instance, in the algebraic sum

The terms are , , and .

Any part of a term that is multiplied by the remaining part is called a **coefficient **of the remaining part. For instance, in the term , the coefficient of is , whereas the coefficient of is . A coefficient such as , which involves no variables, is called a **numerical coefficient**. Terms such as and , which differ only in their numerical coefficients, are called **like terms **or **similar terms**.

An algebraic expression such as can be considered an algebraic sum consisting of just one term. Such a one-termed expression is called a **monomial**. An algebraic sum with two terms is called a **binomial**, and an algebraic sum with three terms is called a **trinomial**. For instance, the expression is a binomial, whereas is a trinomial. An algebraic sum with two or more terms is called a **multinomial**.

A **polynomial **is an algebraic sum, in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers. For instance, the trinomial is not a polynomial; however, the trinomial is a polynomial in the variables and . A term such as , which contains no variables, is called a **constant term **of the polynomial. The numerical coefficients of the terms in a polynomial are called the **coefficients **of the polynomial. The coefficients of the polynomial above are , , and .

The **degree of a term **in a polynomial is the sum of all the exponents of the variables in the term. In adding exponents, you should regard a variable with no exponent as being a first power. For instance, in the polynomial , the term has degree , the term has degree , and the term has degree . The constant term, if it is zero, is always regarded as having degree .

The highest degree of all terms that appear with nonzero coefficients in a polynomial is called the **degree of the polynomial. **For instance, the polynomial considered above has degree . Although the constant monomial is regarded as a polynomial, this particular polynomial is not assigned a degree.

__Example__:

In each case, identify the algebraic expression as a monomial, binomial, trinomial, multinomial, and/or polynomial and specify the variables involved. For any polynomials, gives the degree and coefficients.

**(a) **

**(b) **

**(c) **

__Solution__:

**(a)** Monomial in and (not a polynomial because of the variable in the denominator).

**(b) **Binomial in and , multinomial (not a polynomial because of the negative exponent on).

**(c) **Trinomial, multinomial, polynomial in and of degree with coefficients , , and .

A polynomial of degree in a single variable can be written in **general form**

in which are the numerical coefficients (although any of the other coefficients can be zero), and is the constant term.