Types of Algebraic Expressions: Definitions, Explanations, Examples

Types of Algebraic Expressions: We are very familiar with the arithmetic symbols like \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9.\) These arithmetic symbols are also known as numerals. We were using the basic mathematical operations to do arithmetic calculations. 

Algebra is the generalized form of arithmetic. An algebraic expression is a combination of variables and constants connected by signs of fundamental operations, i.e., \(+, -,  ×\) and \( \div .\) Based on the number of terms, an algebraic expression can be classified into various types. In this article, we will learn about the types of algebraic expressions in detail.

Algebraic Expressions

Variables: In algebra, we come across two types of symbols, namely, constants and variables. A variable is a quantity that may change within the context of a mathematical problem. When a variable is used in a function, we know that it is not just one constant number, but it can represent many numbers. For example, \(a, b, pq, -xy,\) etc.

Constants: A symbol with a fixed numerical value in all situations is called a constant, for instance, \(10, -11, 0.025,\) etc., 
Thus, an algebraic expression is a collection of one or more terms separated by the signs plus or minus.
For example,

Terms of an Algebraic Expression

We have already learnt that a combination of constants and variables connected by the four fundamental operations is called an algebraic expression. For example, \(3x+2y-10\) is an algebraic expression. This expression is formed from variables \(x\) and \(y\) and constants \(3, 2\) and \(10.\) The product of \(3\) and \(x\) is added to the product of \(2\) and \(y,\) and from the sum of \(3x+2y,\) the constant \(10\) is subtracted. 

Consider an algebraic expression \(2x-3y+8.\) We can write it as \(2x+(-3y)+8.\)
In this algebraic expression, parts separated by plus signs are called terms. The expression \(2x-3y+8\) consists of three terms, and they are \(2x, -3y\) and \(8.\)

The numerical part, including the sign of a term, is called the coefficient of the variable. In the above expression, \(2\) is the coefficient of \(x,\) and \(-3\) is the coefficient of \(y.\) The term \(8\) with no variables is called a constant term.

In a term, \(8xy,\) coefficient \(8\) is called the numerical coefficient, and \(xy\) is called the literal coefficient.
Thus, we will find that the expressions we deal with can always be seen this way. They have parts that are formed separately and then added. Such parts of an expression that are formed separately and then added are known as terms.

Like and Unlike Terms

When terms have the same algebraic factors, they are known as like terms, and when they have different algebraic factors, they are known as, unlike terms. For example, in the expression \(3xy-4x+6xy-4,\) look at the terms \(3xy\) and \(6xy.\)

Let us look at more examples of like and unlike terms to understand it better.

Types of Algebraic Expressions

According to the number of terms used to form an algebraic expression, it is called monomial, binomial, trinomial, and so on, as explained below.

Monomial: An algebraic expression having only one term is called a monomial. For example, \(3y,\,\,2xyz,4x, – xy,\frac{{ – 5}}{3}abc,\) etc., are monomials.

Binomial: An algebraic expression that contains two unlike terms is called a binomial. For example, \(x + y,4p + 2z,3{x^2} – {y^2}\) etc., are binomials.

Trinomial: An algebraic expression having three unlike terms is called a trinomial. For example, \(a + 4b + 2z,x – pq + yz,\) etc., are trinomials.

Quadrinomial: An algebraic expression containing four terms is called a quadrinomial. For example, \(7ab – c + z + 4xy,abc – a – b – c\) etc., are quadrinomials.

Multinomial: An algebraic expression with two or more than two terms is called a multinomial. For example, each of \(4x + 3,5 – x,{y^2} + 7y\) is a multinomial of two terms. \(9 + x – y + xy,\) is a multinomial of four terms. \(a – ab + 7b – 6bc – z,\) is a multinomial of five terms and so on.

Polynomial: An algebraic expression with one or more unlike terms with the power of the variables as only whole numbers is called a polynomial. In other words, all monomials, binomials, trinomials and all other expressions having any number of finite terms with the power of their variables as whole numbers are called polynomials. No term in a polynomial contains a negative exponent or any variable in the denominator. For example, \(4{x^3} + 3{x^2} + 9x – 1,9{a^2} + 5,\)etc., are polynomials.

Points to Remember

1. Terms are separated by plus or minus sign only.
2. The signs of multiplication and division do not separate terms.
3. \(4p+5z-7y\) has three terms, whereas \(4p×5z÷7y\) has only one term.

Standard Identities

Some identities involve products of specific kinds of algebraic expressions. They are true for any value of the variable. Such identities are known as standard identities, which are as follows.

1. \({(a + b)^2} = {a^2} + {b^2} + 2ab\)
2. \({(a – b)^2} = {a^2} + {b^2} – 2ab\)
3. \((a + b)(a – b) = {a^2} – {b^2}\)
4. \((x + a)(x + b) = {x^2} + (a + b)x + ab\)

\({(a + b)^2}\) and \({(a – b)^2}\) are called perfect squares. \({a^2} – {b^2}\) is called the difference between two squares.

Types of Algebraic Expressions Examples

Example 1: Classify the following expressions as a monomial, a binomial, and a trinomial:

\(a + b,ab + y + z,pq + q – p,6{x^2} + 4x,8xy + 3y – 2z,9,3ab – 3,x,8p – 9q – 12ab + 4z – 3abc.\)

Solution: We know that expression with one term is known as a monomial, with two terms are known as binomial, and three terms are known as trinomial.

Thus, in the given set of expressions, monomial terms are \(x, 9.\)

Binomial expressions are \(a + b,6{x^2} + 4x,3ab – 3,8p – 9q.\)

Trinomial expressions are \(ab + y + z,pq + q – p,8xy + 3y – 2z, – 12ab + 4z – 3abc.\)

Solved Examples – Types of Algebraic Expressions

Q.1. For each expression given below, state whether it is a monomial, polynomial or trinomial. (a) \(3x×2+4+y\) (b) \(3x + 4 \div \frac{a}{3}\) (c) \(xy÷4\)
Ans: (a) The expression \(3x×2+4+y\) consists of three terms, and thus, is a trinomial.
(b) The expression \(3x + 4 \div \frac{a}{3}\) consists of two terms, and thus, is a binomial.
(c) The expression \(xy÷4\) consists of only one term, and thus, is a monomial.

Q.2. What type of algebraic expression is \(8x+5\)?
Ans: \(8x+5\) has two unlike terms \(8x\) and \(5,\) and hence it is a binomial. Every binomial is a polynomial as well. So \(8x+5\) is a polynomial as well. Therefore, \(8x+5\) is a binomial as well as a polynomial.

Q.3. Sort out the following algebraic expression as monomial, binomial, trinomial.
(a) \(10-b-4abc\) (b) \(3a-5b×2c\)
Ans: (a) \(10-b+c-4abc\) has three terms: \(10, -b,\) and \(-4abc.\) Thus, it is a trinomial.
(b) \(3a-5b×2c\) has only two terms: \(3a,\) and \(-5b×2c.\) Thus, it is a binomial.

Q.4. State the number of terms in each of the following below-given expressions.
(a) \(3a-b\) (b) \(a÷b×c\) (c) \(2×a+b÷c×q-y\)
Ans: (a) \(3a-b\) has two terms, i.e., \(3a\) and \(b.\)
(b) \(a÷b×c\) has only one term, i.e., \(\frac{{ab}}{c}.\)
(c) \(2×a+b÷c×q-y\) has three terms, i.e., \(2a,\frac{b}{{cq}}\) and \(-y.\)

Q.5. State the coefficient of \(x\) and \(y\) in the algebraic expression: \(10x-10y+3.\)
Ans: The numerical part, including the sign or symbol of any term, is the variable’s coefficient. In the given algebraic expression, \(10x-10y+3, x\) and y are variables here, and thus, the coefficient of \(x\) is \(10,\) and the coefficient of \(y\) is \(-10.\)

Summary

In this article, we first started to learn the basics of algebra. We learnt about the variables, constants, algebraic expressions, like terms and unlike terms, etc. Then we learnt that based on the number of terms in an algebraic expression, we could divide them as monomial, binomial, trinomial, polynomial etc.

In addition to this, we also learnt the formulas of the standard identities. And lastly, we learnt to solve some examples to strengthen our grip over the concept.

Frequently Asked Questions (FAQs) – Types of Algebraic Expressions

Q.1. What is an algebraic expression and its types?
Ans: An algebraic expression is a combination of variables and constants connected by signs of fundamental operations, i.e., \(+, -, ×\) and \( \div .\) Based on the number of terms, an algebraic expression can be classified into various types: monomial, binomial, trinomial, quadrinomial, polynomial, etc.

Q.2. How many types of algebraic expressions are there. Explain?
Ans: There are various types of algebraic expressions. They are
Monomial: An algebraic expression having only one term is called a monomial.
Binomial: An algebraic expression that contains two unlike terms is called a binomial.
Trinomial: An algebraic expression having three unlike terms is called a trinomial.
Multinomial: An algebraic expression with two or more than two terms is called a multinomial.
Quadrinomial: An algebraic expression containing four terms is called a quadrinomial.
Polynomial: An algebraic expression with one or more unlike terms with the power of variables as only whole numbers is called a polynomial. In other words, all monomials, binomials, trinomials and all other expressions having any number of finite terms and whole number power for variables are called polynomials.

Q.3. What are the four identities of algebraic expressions?
Ans: The four identities of algebraic expressions are;
1. \({(a + b)^2} = {a^2} + {b^2} + 2ab\)
2. \({(a – b)^2} = {a^2} + {b^2} – 2ab\)
3. \((a + b)(a – b) = {a^2} – {b^2}\)
4. \((x + a)(x + b) = {x^2} + (a + b)x + ab\)

Q.4. How do you classify algebraic expressions?
Ans: Based on the number of terms in an algebraic expression, we can classify them as a monomial, binomial trinomial, polynomial etc. If an expression consists of only one term, then the expression is known as a monomial. If it contains two terms, then binomial. If it contains three terms, then trinomial, etc.

Q.5. Define polynomials.
Ans: All monomials, binomials, trinomials and all the other expressions having any number of finite terms are called polynomials, provided no term contains a negative exponent or any variable in the denominator.

Now you are provided with all the necessary information on the types of algebraic expressions and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.