What is Algebraic Expression(Definition, Formulas & Examples)

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In natural languages such as English, we use sentences and statements for communication. Similarly, the statements used in Mathematics to express a scenario are called ‘expressions’. The expressions containing unknown quantities(or variables) along with other components like constants and operators are called algebraic expressions. For example, $3x + 2$ is an algebraic expression where the terms $3$ and $2$ are constant and the symbol $x$ is a variable separated by the arithmetic operation + (plus).

Let’s understand what is an algebraic expression and the algebraic expression formula and terms associated with them.

Algebraic Expression Formula

The way of representing algebraic expressions is called the algebraic expression formula. It is a conventional way of writing algebraic expressions. 

What is Algebraic Expression in Math?

An algebraic expression is a mathematical statement consisting of variables, constants(numbers), and arithmetic operations between them. 

Algebraic expressions are the way of expressing a situation using letters or alphabets without specifying their actual values. These letters are called here as variables. An algebraic expression can be a combination of both variables and constants.

Algebraic Expression Examples

Following are some examples of algebraic expressions.

Parts of an Algebraic Expression

An algebraic expression is made up of the following components

• Variables: The unknowns in algebraic expressions represented by letters such as $x$, $y$, $z$, $a$, $b$, $c$, etc. 
• Constants: The numbers without a variable part in algebraic expressions.
• Coefficients: The number multiplied by a variable in algebraic expressions.
• Arithmetic Operators: The mathematical operators $+$(plus), $-$(minus), $\times$(multiplication) and $\div$(division)

Let us consider an expression, $2x + 5y – 7$. 

Here, the parts of the expression are:

• Variables are $x$ and $y$
• Constant is $7$
• Coefficients are $2$ and $5$
• Mathematical operators used are plus (+) and minus (-).

Now, let us understand what are terms, factors, and coefficients in an algebraic expression in detail.

What is a Term in an Algebraic Expression?

A term can be a number, a variable, a product of two or more variables, or a product of a number and a variable. An algebraic expression is formed by a single term or by a group of terms.

For example, in the expression $2x + y$, the two terms are $2x$ and $y$.

The terms add up to form the expression.

Let’s consider a few examples to understand what is a term in an algebraic expression.

In an algebraic expression $x + 6xy – 4y + 7$, the terms are

• $x$: a single variable without any number part
• $6xy$: a product of a number $6$ and two different variables $x$ and $y$. Here $6xy = 6 \times x \times y$
• $-4y$: a product of a number $-4$ and a variable $y$. Here $-4y = -4 \times y$
• $7$: a single number without any variable part

On adding these terms together, $\left(x \right) + \left(6xy \right) + \left(-4y \right) + \left(7 \right)$, we get $x + 6xy – 4y + 7$, which is an algebraic expression. 

What is a Factor of a Term?

The numbers or variables that are multiplied to form a term are called its factors. For example, $6xy$ is a term of algebraic expression $x + 6xy – 4y + 7$ and $6$, $x$ and $y$ are the factors of a term $6xy$.

Similarly, factors of the term $-4y$ are $-4$ and $y$.

Although $x$ can be written as $1 \times x$ and $7$ can be written as $1 \times 7$, $1$ is not taken as a separate factor.

Let’s consider, another algebraic expression, $2a^{3} + 3a^{2}b$.

Here, the terms are $2a^{3}$ and $3a^{2}b$.

And, the factors of $2a^{3}$ are $2$, $a$, $a$ and $a$, since,  $2a^{3} = 2 \times a \times a \times a$.

Similarly, the factors of $3a^{2}b$ are $3$, $a$, $a$ and $b$. Here also, $3a^{2}b = 3 \times a \times a \times b$.

What is a Coefficient in an Algebraic Expression?

A coefficient is an integer that is written along with a variable or it is multiplied by the variable.  In other words, a coefficient is the numerical factor of a term containing constants and variables. 

For example, in the term $5y$, $5$ is the coefficient.

Similarly, $-2$ is a coefficient of the term $-2a^{2}b$.

When there is no numerical factor in a term, its coefficient is taken as $1$. For example, in the term $x^3$, the coefficient is $1$.

Similarly, in the term $-y$, the coefficient is $-1$.

Types of Terms – Like and Unlike Terms in Algebra

There are two types of algebraic expression terms. These are 

• Like Terms 
• Unlike Terms

What are Like Terms in Algebra?

The algebraic terms whose variables as well as exponents are the same are called like terms. The terms that have only the same variables are not treated as like terms. Similarly, the terms whose exponents are the same are also not considered like terms.

Examples of like terms are

$5x^{2}$ and $-3x^{2}$ (Both have the same terms $x^{2}$)

$-3ab$ and $-2ab$ (Both have the same terms $ab^{2}$)

$8m^{2}n$ and $m^{2}n$ (Both have the same terms $m^{2}n$)

Note: Coefficients are not considered while checking for like terms.

What are Unlike Terms in Algebra?

The algebraic terms that are not like terms are called, unlike terms. Two algebraic terms are unlike terms if they have different variables or the same variables with different exponents.

Examples of unlike terms are

$3a$ and $3b$ (Variable of first term is $a$ and that of second term is $b$)

$2m^{2}$ and $5m$ (Variables in both the terms are the same $m$, but the exponents are different)

Types of Algebraic Expressions

The type of an algebraic expression is based on the variables found in that particular expression, the number of the terms of that expression, and the values of the exponents of the variables in each expression. Based on this classification an algebraic expression can be either of the following.

• Monomial
• Binomial
• Trinomial
• Polynomial
• Multinomial

Monomial

An algebraic expression that consists of one non-zero term only is called a monomial.

Following are examples of monomials.

• $8x$ is a monomial in one variable $x$
• $7a^{2}b$ is a monomial in two variables $a$ and $b$
• $-3xy$ is a monomial in two variables $x$ and $y$
• $m^{3}$ is a monomial in one variable $m$

Binomial

An algebraic expression that consists of two terms is called a binomial.

Following are examples of binomials.

• $8x + 3y$ is a binomial in two variables $x$ and $y$
• $3a^{2}b – 3ab^{2}$ is a binomial in two variables $a$ and $b$

Trinomial

An algebraic expression that consists of three terms is called a trinomial.

Following are examples of trinomials.

• $x + y + z$ is a trinomial in three variables $x$ , $y$ and $z$
• $l^{2} – 2lm + m^{2}$ is a trinomial in two variables $l$ and $m$

Polynomial

In general, a word with a variable’s non-negative (or positive) integral exponents is defined as a polynomial. In other words, any algebraic expression with one or more terms is called a polynomial.

Following are examples of polynomials.

• $8x$ is a polynomial in one variable $x$. It is also a monomial in one variable $x$.
• $3a^{2}b – 3ab^{2}$ is a polynomial in two variables $a$ and $b$. It is also a binomial in two variables $a$ and $b$.
• $l^{2} – 2lm + m^{2}$ is a polynomial in two variables $l$ and $m$. It is also a trinomial in two variables $l$ and $m$.
• $x^{3} – 3x^{2}y + 3xy^{2} – y^{3}$ is a polynomial in two variables $x$ and $y$.
• $a^{3} – 3b^{2} + 2ac + 5d – 4abcd$ is a polynomial in four variables $a$, $b$, $c$ and $d$.

Multinomial

An algebraic expression with one or more terms (the exponents of variables can be either positive or negative) is called a multinomial. 

The following are examples of multinomials.

• $8x$ is a multinomial in one variable $x$. It is also a monomial and polynomial in one variable $x$.
• $3a^{2}b – 3ab^{2}$ is a multinomial in two variables $a$ and $b$. It is also a binomial and polynomial in two variables $a$ and $b$.
• $l^{2} – 2lm + m^{2}$ is a multinomial in two variables $l$ and $m$. It is also a trinomial and polynomial in two variables $l$ and $m$.
• $x^{3} – 3x^{2}y + 3xy^{2} – y^{3}$ is a multinomial in two variables $x$ and $y$.
• $a^{3} – 3b^{2} + 2ac + 5d – 4abcd$ is a multinomial in four variables $a$, $b$, $c$ and $d$.
• $a^{2} + 2ab + b^{-2}$ is a multinomial in two variables $a$ and $b$ but not a polynomial. (It has a negative or a fractional exponent for variable $b$)
• $x^{\frac{1}{2}} + 5x + 7$ is a multinomial in one variable $x$ but not a polynomial. (It has non-integral (fractional) exponent for variable $x$)
• $2x^{0.25} – 4xy + 8y$ is a multinomial in two variables $x$ and $y$ but not a polynomial. (It has non-integral (decimal) exponent for variable $x$)

Note:

• All polynomials are multinomials
• All multinomials are not polynomials. (Multinomials with negative exponents of variables are not polynomials)

Practice Problems

1. Identify variables, constants, coefficients and arithmetic operators in the following algebraic expressions.
   • $x + y – z$
   • $2a^{2} – 3ab + 7b^{2}$
   • $x^{3} + \frac{2}{7}y^{2} + 8$
   • $a^{3} – 3a^{b} + 3ab^{2} – b^{3}$
   • $\frac{7l^{3} + 2lm – 8lm^{2}}{9}$
2. How many terms are there in the following algebraic expressions?
   • $2x$
   • $2 + x$
   • $2 – x$
   • $\frac{2}{x}$
   • $5x^{2} + 2x + 7 + 3x$
   • $5xyz$
3. State True or False
   • All monomials are polynomials
   • All polynomials are binomials
   • All trinomials are polynomials
   • All binomials are multinomials
   • All multinomials are polynomials
   • All polynomials are multinomials

FAQs

Conclusion

An algebraic expression is a mathematical statement consisting of variables, constants(numbers), and arithmetic operations between them. Depending on the number of terms and the nature of exponents of variables in algebraic expressions, they are classified as monomials, binomials, trinomials, polynomials, or multinomials.

Recommended Reading

• Addition of Algebraic Expressions(With Methods & Examples)
• What is Algebra – Definition, Basics & Examples
• What is Pattern in Math (Definition, Types & Examples)
• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)