Expressions with variables

Algebra is the branch of mathematics that deals with symbols and variables. Alphabetical letters are used to find unknown numbers from the equations. Algebra is divided into sub-branches such as elementary algebra, advanced algebra, linear algebra, and commutative algebra. It covers algebraic expressions, formulas, and identities, which are used in solving many mathematical problems. An algebraic expression is an expression consisting of variables, constants, coefficients, and mathematical operations like addition, subtraction, etc. An algebraic expression is a form of writing equations using letters or alphabets without specifying their values. These letters are called variables. In this article, we will learn about expressions with variables.

Parts of Variable Expressions

A variable expression is a combination of different terms involving variables, constants, and mathematical operations like addition, subtraction, etc.

Some of the important parts of an algebraic expression are defined below:

1. Variable: A symbol without a fixed value is called a variable. It can take any value. It is represented by alphabetical letters like x, y, z, etc. 

2. Constant: A symbol with a fixed numerical value is called a constant.

3. Term: A term is a variable or a constant alone or a combination of both combined by mathematical operations.

4. Coefficients: The quantity multiplied by a variable and remains constant throughout the problem is known as a coefficient. 

Algebraic expression example: 9x + 7.

Here x is the variable, 9 is the coefficient 9x, 7 is the constant, and 9x and 7 are the two terms in the expression. 

Types of Algebraic Expressions

There are three main types of algebraic expressions based on the number of terms:

• Monomial expression:  An algebraic expression with only one term is known as a monomial expression.

For example – 3x, 4, 8y, etc

• Binomial expression: An algebraic expression with only two terms is known as a binomial expression.

For example – $ax + by$, \[x^2+y\], etc

• Trinomial expression: An algebraic expression with three terms is known as a trinomial expression.

For Example: $7x+4y-3$, \[x^2+3y -8\], etc.

• Polynomial expression: An algebraic expression having one variable and the exponent of the variable is a whole number, is known as a polynomial expression.

For example – $ax + bx + cx+6$, \[x^4+ x^2 – x + 4\] etc

Use of Algebraic Expressions

Some of the uses of algebraic expressions are listed below:

• They are used to solve different and complex equations in mathematics.

• They can also be seen in computer programming.

• They are also used in economics to find out the revenue, cost, etc.

• Different branches of mathematics like trigonometry, geometry, etc also use algebraic expressions to find the unknown values of angles.

• Algebraic expressions are also used to represent real-life problems.

Algebraic Expression Formulas

The basic identities used in algebra are also known as algebraic expression formulas:

• \[(a+b)^2=a^2+2ab+b^2\]

• \[(a−b)^2=a^2−2ab+b^2\]

• \[(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc\]

• \[a^2−b^2=(a+b)(a−b)\]

• \[(a+b)^3=a^2+3a^2b+3ab^2+b^3\];\[(a+b)^3=a^3+b^3+3ab(a+b)\]

• \[(a−b)^3=a^3−3a^2b+3ab^2−b^3; (a−b)^3=a^3−b^3−3ab(a-b)\]

• \[a^3+b^3=(a+b)(a^2−ab+b^2)\]

• \[a^3−b^3=(a−b)(a^2+ab+b^2)\]

Interesting facts

• The terms in algebraic expressions that are constants or involve the same variables raised to the same exponents are called like terms. For example: \[6x^2 + x – 4x^2 + 9\], here \[6x^2\] and \[– 4x^2\] are similar terms.

• The terms in algebraic expressions that do not have the same variables or have the same variables but are raised to different exponents are called, unlike terms. For example: \[6x^4 + x – 4x^2 + 9\], here \[6x^2\], x and 9 are the unlike terms.

• All the polynomials are algebraic expressions but all algebraic expressions are not polynomials.

• The algebraic expressions which do not have fractional or non–negative exponents are polynomials.

Solved questions

Q1. Name the type of algebraic expression:

1.  2x + 3y + 24xy 

2.  – 3x + 2y 

3. – 20xy

Ans. a. Trinomial b. Binomial c. Monomial

Q2. Evaluate the expression \[6ab + 2b^2 – c\], where a = 2, b = 3, c = 1.

Ans. Put the values of a, b and c is the expression 

\[6(2)(3) + 2(3)^2 – 1 = 53\] 

Q3. Evaluate \[{(a + b)}^{2}\] and verify the identity for a = 2 and b = 1.

Ans.  We know,\[(a+b)^2=a^2+2ab+b^2\]

LHS \[= (a+b)^2 = (2+1)^2 = 9\]

RHS \[= a^2+2ab+b^2 = 2^2 + 2(2)(1) + 1^2 = 4 + 4 +1 = 9\]

Summary

Algebra is the branch of mathematics that deals with symbols and variables. A variable expression is a combination of different terms involving variables, constants, and mathematical operations like addition, subtraction, etc. There are three main types of algebraic expressions based on the number of terms: Monomial, Binomial, and Polynomial.

Practice questions

Q1. Name the type of algebraic expression:

1. \[{2x}^{2} \]

2.  – 3x + y 

3. \[– 20xy + 3 – {2y}^2\]

Ans. a. monomial b. binomial c. trinomial

Q2. Evaluate the expression ab + 2b – c, where a = 1, b = 2, c = 3.

Ans. 3

Q3. Evaluate ${(a-b)}^2$ and verify the identity for a = 1 and b = - 1.

Ans. 4

List of related articles

• Variables and constants

• Algebra 

• List of algebraic identities

• Basics of algebra