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

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

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

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.