# Introduction to Algebraic Expressions and Polynomials

An **algebraic expression **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, $$7$$,$$x$$, $$2x – 3y + 1$$, $$\frac{{5{x^3} – 1}}{{4xy + 1}}$$, $$\pi {r^2}$$, and $$\pi r\sqrt {{r^2} + {h^2}} $$ 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 $$\frac{{2x – 3y + 1}}{{x{y^2}}}$$ when $$x = 2$$ and $$y = – 1$$.

__Solution__:

Substituting $$x = 2$$ and $$y = – 1$$, we obtain

\[\frac{{2(2) – 3( – 1) + 1}}{{2{{( – 1)}^2}}} = \frac{{4 + 3 + 1}}{{2(1)}} = \frac{8}{2} = 4\]

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

\[3{x^2}y – \frac{{4x{z^2}}}{y} + \pi {x^{ – 1}}y\]

the terms are $$3{x^2}y$$, $$ – 4x{z^2}/y$$, and $$\pi {x^{ – 1}}y$$.

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 $$ – 4x{z^2}/y$$, the coefficient of $${z^2}/y$$ is $$ – 4x$$, whereas the coefficient of $$x{z^2}/y$$ is $$ – 4$$. A coefficient such as $$ – 4$$, which involves no variables, is called a **numerical coefficient**. Terms such as $$5{x^2}y$$ and $$ – 12{x^2}y$$, which differ only in their numerical coefficients, are called **like terms **or **similar terms**.

An algebraic expression such as $$4\pi {r^2}$$ 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 $$3{x^2} + 2xy$$ is a binomial, whereas $$ – 2x{y^{ – 1}} + 3\sqrt x – 4$$ 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 $$ – 2x{y^{ – 1}} + 3\sqrt x – 4$$ is not a polynomial; however, the trinomial $$3{x^2}{y^4} + \sqrt 2 xy – \frac{1}{2}$$ is a polynomial in the variables $$x$$ and $$y$$. A term such as $$ – 1/2$$, 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 $$3$$, $$\sqrt 2 $$, and $$ – 1/2$$.

The **degree of a term **in a polynomial is the sum of all the exponents of the variables in the term. When adding exponents, you should regard a variable with no exponent as being a first power. For instance, in the polynomial $$9x{y^7} – 12{x^3}y{z^2} + 3x – 2$$, the term $$9x{y^7}$$ has degree $$1 + 7 = 8$$, the term $$ – 12{x^3}y{z^2}$$ has degree $$3 + 1 + 2 = 6$$, and the term $$3x$$ has degree $$1$$. The constant term, if it is zero, is always regarded as having degree $$0$$.

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 $$8$$. Although the constant monomial $$0$$ 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, give the degree and coefficients.

**(a) **$$\frac{{4x}}{y}$$

**(b) **$$4x + 3{y^{ – 1}}$$

**(c) **$$ – \frac{5}{3}{x^2}y + 8xy – 11$$

__Solution__:

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

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

**(c) **Trinomial, multinomial, polynomial in $$x$$ and $$y$$ of degree $$3$$ with coefficients $$ – 5/3$$, $$8$$, and $$ – 11$$.

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

\[{a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + \cdots + {a_2}{x^2} + {a_1}x + {a_0}\]

in which $${a_n},{a_{n – 1}}, \ldots ,{a_2},{a_1},{a_0}$$ are the numerical coefficients $${a_n} \ne 0$$ (although any of the other coefficients can be zero), and $${a_0}$$ is the constant term.