• NEET
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • Class 6-10
      • Class 6th
      • Class 7th
      • Class 8th
      • Class 9th
      • Class 10th
    • View All Options
      • Online Courses
      • Offline Courses
      • Distance Learning
      • Hindi Medium Courses
      • International Olympiad
    • NEET
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE (Main+Advanced)
      • Class 11th
      • Class 12th
      • Class 12th Plus
    • JEE Main
      • Class 11th
      • Class 12th
      • Class 12th Plus
  • NEW
    • JEE MAIN 2025
    • NEET
      • 2024
      • 2023
      • 2022
    • Class 6-10
    • JEE Main
      • Previous Year Papers
      • Sample Papers
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • JEE Advanced
      • Previous Year Papers
      • Sample Papers
      • Mock Test
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • NEET
      • Previous Year Papers
      • Sample Papers
      • Mock Test
      • Result
      • Analysis
      • Syllabus
      • Exam Date
    • NCERT Solutions
      • Class 6
      • Class 7
      • Class 8
      • Class 9
      • Class 10
      • Class 11
      • Class 12
    • CBSE
      • Notes
      • Sample Papers
      • Question Papers
    • Olympiad
      • NSO
      • IMO
      • NMTC
    • ALLEN e-Store
    • AOSAT
    • ALLEN for Schools
    • About ALLEN
    • Blogs
    • News
    • Careers
    • Request a call back
    • Book home demo
NCERT SolutionsCBSE NotesCBSE Exam
Home
Maths
Algebraic Expression

Algebraic Expressions

Terms related to algebraic expression Constant : A quantity which has a fixed value, i.e., whose value does not change is called a constant. E.g., 8,−7,0,687​. etc. are all constants.

Variable: A symbol which can be assigned different numerical values is called a variable. In algebra, the variables are denoted by the letters in the English alphabet, viz.; x,y,z,….a,b,c etc. E.g., The perimeter P of a square of a side S is given by the formula, P=4×S, here 4 is a constant while P and S are variables.

1.0Algebraic Expression

A combination of constants and variables connected by signs of fundamental operations (+,−,× and ÷ ) is called an algebraic expression. E.g., 5−3x+4x2y is an algebraic expression consisting of three terms, namely 5,−3x and 4x2y. Expression can also be obtained by multiplying variable by variable. x×x=x2 and x×x×x=x3 Factors: Each term of an algebraic expression consists of a product of constants and variables. A constant factor is called a numerical factor, while a variable factor is known as literal factor. E.g., In the expression 3x2−4xy+7y2;3x2,−4xy,7y2 are all terms of the above expression. Each term is further made of factors, i.e., 3x2 can be written as 3×x×x,−4xy as −4×x×y and 7y2 as 7×y×y. Here 3,x,x are factors of 3x2, −4,x,y are factors of −4xy and 7,y,y are factors of 7y2. Thus, an expression, its terms and factors of the terms can be represented by a tree diagram to make it easily comprehensible to you.

Factors are numbers that can't be further factorized >1 is not taken as separate factor.

Coefficients

In a term, coefficient is either a numerical factor, an algebraic factor or the product of two or more factors.

E.g., In the algebraic expression 10xy,10 is the coefficient of xy,10x is the coefficient of y and 10 y is the coefficient of x. The numerical part is called the numerical coefficient and literal part or variable or variable part is called the literal coefficient. When the numerical coefficient is not given. It is always understood to be 1 . E.g.,

Term

Numerical Coefficient

Literal Coefficient

84x2z84x2z
−32xy-32xy

Always consider the "sign" while taking the coefficient. E.g., In −7x2 coefficient of y2 is −7x and numerical coefficient is -7 not ' 7 '.

Like and Unlike Terms

Like terms:

  • Like terms or similar terms are terms which have the same literal factor. They may differ in their numerical coefficient. E.g., (i) 9x2,5x2 and −2x2 are like terms. (ii) −6x2y and 4yx2 are like terms.

Unlike terms :

  • Unlike terms are terms which have different literal factors. E.g., 7x2 and 9y2, the literal factors are x2 in 7x2 and y2 in 9y2, which are different.

2.0Various Types of Algebraic Expressions

Monomial: An algebraic expression which contains only one term, is called a monomial. Thus, 5x,2xy,−3a2b,−7, etc, are all monomials.

Binomial : An algebraic expression containing two unlike terms is called a binomial. Thus, (2a+3b),(8−3x),(x2−4xy2) etc, are all binomials.

Trinomial : An algebraic expression containing three terms is called a trinomial. Thus, (a+2b+5c),(x+2y−3z),(x3−y3−z3), etc, are all trinomials.

Quadrinomial : An algebraic expression containing four terms is called a quadrinomial. Thus, (x+y+z−5),(x3+y3+z3+3xyz), etc, are all quadrinomials.

Polynomial : An expression containing one or more terms is called a polynomial. An algebraic expression of the form a+bx+cx2+dx3+…. , where a,b,c,d are constants and x is a variable is called a polynomial in x . The degree of the polynomial is the greatest power of the variable present in the polynomial. In any polynomial, the power of variable is a non-negative integer. E.g., 9x4−29​x2−57​x−2 is a polynomial in x of degree 4 . If the polynomial is in two or more variables, then the sum of the powers of the variables in each term is taken and the greatest sum is the degree of the polynomial. Note : Every polynomial is an expression, but every expression need not be a polynomial.

3.0Operations On Algebraic Expressions

Addition Of Algebraic Expressions

While adding algebraic expressions, we collect the like terms and add them. The sum of several like terms is another like term whose coefficient is the sum of the coefficients of those like terms.

Subtraction Of Algebraic Expressions

The difference of two like terms is a like term whose coefficient is the difference of the numerical coefficients of the two like terms.

Rule for subtraction of algebraic expressions: Change the sign of each term of the expression to be subtracted and then add.

Note: At the time of adding or subtracting the expression, arrange the expressions so that the like terms are grouped together or arrange the expression in rows so that like terms appear in columns and then add or subtract.

If the operation of subtraction is performed by writing in column form, then the sign of each term to be subtracted must be changed and then the terms are added or subtracted.

  • While adding or subtracting algebraic expressions, like terms will be added or subtracted to like terms only.

Multiplication Of Algebraic Expressions

Before taking up the product of algebraic expressions, let us look at two simple rules. (i) The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative. (ii) If a is any variable and m,n are positive integers then am×an=(am+n)

Thus, x3.x5=x3+5=x8 and x7+1=x8, etc.

Multiplication of monomials - Rules:

(i) The coefficient of the product of two monomials is equal to the product of their coefficients.

(ii) The variable part in the product of two monomials is equal to the product of the variables in the given monomials. These rules may be extended for the product of three or more monomials.

Volume of cuboid with length ( ℓ ), breadth (b) and height ( h ) is lbh cubic units. If a trinomial is a perfect square, then it is the square of the binomial.

Multiplication of a monomial and a binomial

Let p,q and r be three monomials. Then, by distributive law of multiplication over addition, we have p×(q+r)=(p×q)+(p×r)

  • x×x=x2
  • x+x=2x

Multiplication of two Binomials

Suppose (a+b) and (c+d) are two binomials. By using the distributive law of multiplication over addition twice, we may find their product as given below: (a+b)×(c+d)=a×(c+d)+b×(c+d)=(a×c+a×d)+(b×c+b×d)=ac+ad+bc+bd. This method is known as the horizontal method.

Division Of Algebraic Expressions

Division of Monomials

To divide one algebraic term by another, the power or exponent rule for division is used. (i) The power of all the factors in the denominator is subtracted from the power of like factors in the numerator. (ii) The numerical coefficient of the numerator and denominator is divided by their HCF.

If either, or both terms are fractions, then the division sign is changed to multiplication and the second term (fraction) is inverted. Then division is then carried out as explained in (i) above or simplify by reducing to the lowest terms.

  • Division by 0 is not defined. 0x​= not defined

4.0Value of an expression

The value of an expression can be found by substituting the given value of literal in the expression. Eg. value of x2+2x+1 when x=5 is 52+2×5+1=25+10+1=36.

5.0Formulas and Rules Using Algebraic Expressions

Perimeter formulas

(i) Perimeter of an equilateral triangle =3ℓ, where ℓ is the length of the side of the equilateral triangle. (ii) Perimeter of a square =4ℓ, where ℓ is the length of the side of the square. (iii) Perimeter of a regular pentagon =5ℓ, where ℓ is the length of the side of the pentagon.

Area formulas

(i) Area of a square =ℓ2, where ℓ is the side of the square. (ii) Area of a rectangle =ℓ×b, where ℓ and b are respectively the length and the breadth of the rectangle. (iii) Area of triangle =2b×h​, where b is the base of the triangle and h is the height of the triangle.

6.0Rules For Number Patterns

(i) If a natural number is denoted by n , its successor is (n+1). (ii) If a natural number is denoted by n,2n is an even number and (2n+1) is an odd number.

Make similar pattern with basic figures as shown:

(The number of line segments required to make the figure is given to the right. Also, the expression for the number of line segments required to make n letters is also given).

7.0Numerical Ability

Q. Draw a tree diagram of the expression 4x2+5y−8z+1. Answer:

Q. Write down the numerical as well as the literal coefficient of each of the following expression. (a) 7x2y2z (b) −28x2yzq2 Solution:

Expression

Numerical Coefficient

Literal Coefficient

(a) 7x2y2z7x2y2z
(b) −28x2yzq2−28x2yzq2

Q. Identify the like terms in each of the following: (a) 9x2,8xy,−2xy,25​xy (b) 8m2n2,−mp2,−m2n2,nm2 Solution: (a) Like terms: 8xy,−2xy,25​xy (b) Like terms: 8 m2n2,−m2n2

Q. Find the coefficient of x in the following: (a) x (b) −3xy2 (c) 4p2x Solution: (a) 1 (b) −3y2 (c) 4p2

Q. Which of the following is a polynomial? (a) 7a2−3ab+5b2+9 (b) b29​+a+ab Solution: (a) 7a2−3ab+5b2+9 is a polynomial in a and b. The sum of the powers of variables of each term is 2,2,2 respectively. (b) b29​+a+ab is not a polynomial in a and b. The sum of the powers of variables of each term is −2,1,2 respectively.

Q. State the degree of the polynomial: a2b2−ab+3ab2. Solution: a2b2−ab+3ab2 is a polynomial in a and b. The sum of the powers of variables of each term is 4,2,3 respectively. ∴ The degree is highest power of variable of a term in the expression. Here, highest power is 4 . So, it is a polynomial of degree 4 .

Q. Classify the following polynomials as monomial, binomial or trinomial? 3x+4y+z,3x2−2x2,x+3y−2x,4p2−2z2−7z2−p2 Solution: 3x+4y+z has 3 terms, so it is a trinomial. 3x2−2x2=x2 has 1 term, so it is a monomial. x+3y−2x=3y−x has 2 terms, so it is a binomial. 4p2−2z2−7z2−p2=3p2−9z2 has 2 terms, so it is a binomial.

Q. Add: 5x2−7x+3,−8x2+2x−5 and 7x2−x−2 Solution: Required sum =(5x2−7x+3)+(−8x2+2x−5)+(7x2−x−2) =5x2−8x2+7x2−7x+2x−x+3−5−2 [collecting like terms] =(5−8+7)x2+(−7+2−1)x+(3−5−2)quad[ adding like terms] =4x2−6x−4

Q. Add : (3x2−51​x+37​)+(−41​x2+31​x−61​)+(−2x2−21​x+5) Solution: Required sum =(3x2−51​x+37​)+(−41​x2+31​x−61​)+(−2x2−21​x+5) =3x2−41​x2−2x2−51​x+31​x−21​x+37​−61​+5 [collecting like terms] =(3−41​−2)x2+(−51​+31​−21​)x+(37​−61​+5) [adding like terms] =(412−1−8​)x2+(30−6+10−15​)x+(614−1+30​) =43​x2−3011​x+643​

Q. Subtract (2x2−5x+7) from (3x2+4x−6) Solution: We have, (3x2+4x−6)−(2x2−5x+7) =3x2+4x−6−2x2+5x−7 =(3−2)x2+(4+5)x+(−6−7) =x2+9x−13

Q. Take away (58​x2−32​x3+23​x−1) from (5x3​−23​x2+32​x+41​) Solution: We have, (5x3​−23​x2+32​x+41​)−(58​x2−32​x3+23​x−1) =5x3​−23​x2+32​x+41​−58​x2+32​x3−23​x+1 =(51​+32​)x3+(−23​−58​)x2+(32​−23​)x+(41​+1) =15(3+10)​x3+(10−15−16​)x2+6(4−9)​x+4(1+4)​ =1513​x3−1031​x2−65​x+45​

Q. −8ab2c,3a2b and −61​ Solution: (−8ab2c)×(3a2b)×(−61​)

=(−8×3×6−1​)×(a×a2×b2×b×c)=4a(1+2)×b(2+1)×c=4a3b3c

Q. Multiply 85​a3b2,12a2b and 6c Solution: (85​a3b2)×(12a2b)×(6c)

=(85​×12×6)×(a3×a2×b2×b×c)=45×a(3+2)×b(2+1)×c=45a5 b3c.

Q. Multiply 4,125​x and −8x2y Solution: 4×(125​x)×(−8x2y)

=[4×125​×(−8)]×(x×x2×y)=3−40​×x(1+2)×y=3−40​x3y

Q. Multiply −a,a2bc and 5−2​a2c2 Solution: (−a)×(a2bc)×(5−2​ab2c2)

=(−1×5−2​)×a×a2×a×b×b2×c×c2=52​×a(1+2+1)×b(1+2)×c(1+2)=52​a4b3c3.

Q. Multiply (−5x2y),(3−2​xy2z),(158​xyz2) and (4−1​z) Verify the result when x=1,y=2 and z=3. Solution: We have (−5x2y)×(3−2​xy2z)×(158​xyz2)×(4−1​z) =(−5×3−2​×158​×4−1​)×(x2×x×x×y×y2×y×z×z2×z) =9−4​×x(2+1+1)×y(1+2+1)×z(1+2+1)=9−4​x4y4z4.

  • Verify: (−5x2y)=−5×(1)2×(2)=−10 (3−2​xy2z)=3−2​×(1)×(2)2×3=−8 (158​xyz2)=158​×1×2×32=158​×2×9=548​ (−41​z)=−41​×3=4−3​ (−5x2y)×(3−2​xy2z)×(158​xyz2)×(−41​z) (−10)×(−8)×(548​)×(−43​)=−576 9−4​x4y4z4=9−4​(1)4×(2)4×(3)4=9−4​×16×81=−576 Hence (−5x2y)×(3−2​xy2z)×(158​xyz2)×(−41​z)=(−94​)x4y4z4

Q. Multiply : 29​x2y by (x+2y). Solution: We have 29​x2y×(x+2y)=(29​x2y×x)+(29​x2y×2y) [by distributive law] =(29​×x2×x×y)+(29​×2×x2×y×y) ={29​x(2+1)×y}+{9×x2×y(1+1)} =29​x3y+9x2y2.

Q. Multiply: (3x−54​y2x) by 21​xy. Solution: We have (3x−54​y2x)×21​xy=(3x×21​xy)−(54​y2x×21​xy) =(3×21​×x×x×y)−(54​×21​×y2×y×x×x) ={23​×x(1+1)×y}−{52​×y(2+1)×x(1+1)}=(23​x2y)−(52​y3x2). column method we have 3x−54​y2x ×21​xy 23​x2y−52​y3x2

Q. Multiply (51​x−41​y) and (5x2−4y2). Solution: We have: Using horizontal method, we have (51​x−41​y)×(5x2−4y2)=51​x(5x2−4y2)−41​y(5x2−4y2) =(51​x×5x2)−(51​x×4y2)−(41​y×5x2)+(41​y×4y2) =x3−54​xy2−45​x2y+y3

Q. Multiply (5x2−6x+9) by (2x−3). Solution: By column method, we have

5x2−6x+9×2x−3​​10x3−12x2+18x

multiplication by 2 x

−15x2+18x−27

multiplication by - 3 Add: 10x3−27x2+36x−27 [multiplication by (2x−3)] ∴(5x2−6x+9)×(2x−3)=10x3−27x2+36x−27.

Q. 4a÷a Solution: 4a÷a=a4a​=4a1−1=4a0=4

Q. 4a2b÷ab Solution: 4a2 b÷ab=ab4a2 b​=4a2−1 b1−1=4a

Q. 9a4b2÷3a2b Solution: 9a4b2÷3a2b=3a2b9a4b2​=3a4−2b2−1=3a2b

Q. If p=−2, find the value of: (i) 4p+8 (ii) −3p2+5p+7 (iii) −2p3−3p2+4p+7 Solution: (i) Value of (4p+8), when p=−2 is 4(−2)+8=−8+8=0 (ii) Value of (−3p2+5p+7) when p=−2 is −3(−2)2+5(−2)+7 =−3×4−10+7=−12−10+7=−15 (iii) Value of (−2p3−3p2+4p+7) when p=−2 is −2(−2)3−3(−2)2+4(−2)+7 =16−12−8+7=3

Q. If a=1,b=−1, find the value of: (i) a2+b2 (a−b)2=a2+b2−2ab (ii) a2+ab+b2 (iii) a2−b2 Solution: (i) Value of a2+b2, when a=1,b=−1 is (1)2+(−1)2=1+1=2 (ii) Value of a2+ab+b2, when a=1,b=−1 is (1)2+(1)(−1)+(−1)2=1−1+1=1 (iii) Value of a2−b2, when a=1, b=−1 is (1)2−(−1)2=1−1=0

Q. Find the value of ' a ' that will make the expression 3x2−8x+ a equal to 5 , at x=2. Solution: Value of 3x2−8x+a, when x=2 is 3(2)2−8(2)+a=12−16+a=a−4 Now, a−4=5 ⇒a=9

Q. Find the value of m if the expression x4−5x3+mx−4 equals -1 when x=−1. Solution: The value of (x4−5x3+mx−4) is -1 , when x=−1 ∴(−1)4−5(−1)3+m(−1)−4=−1 ⇒1+5−m−4=−1 ⇒2−m=−1 ⇒−m=−1−2=−3 ⇒m=3

Q. The side of a square is 5cm. Find its perimeter and area. Solution: Side of square (ℓ)=5 cm Perimeter =4(ℓ)=4×5=20 cm Area =ℓ2=5×5=25 cm2

Q. If the perimeter of square is 36 cm . Find its area. Solution: Perimeter of square =36 cm Side (ℓ)=436​ cm=9 cm Area of square =ℓ2 =(9)2=81 cm2

Q. If W=XP+2r, then X is equal to Solution: XP=W−2r X=PW−2r​

Table of Contents


  • 1.0Algebraic Expression
  • 1.1Coefficients
  • 1.2Like and Unlike Terms
  • 2.0Various Types of Algebraic Expressions
  • 3.0Operations On Algebraic Expressions
  • 3.1Addition Of Algebraic Expressions
  • 3.2Subtraction Of Algebraic Expressions
  • 3.3Multiplication Of Algebraic Expressions
  • 3.4Division Of Algebraic Expressions
  • 4.0Value of an expression
  • 5.0Formulas and Rules Using Algebraic Expressions
  • 5.1Perimeter formulas
  • 5.2Area formulas
  • 6.0Rules For Number Patterns
  • 7.0Numerical Ability

Related Article:-

Algebraic Expressions

An expression, its terms and factors of the terms can be represented by a tree diagram to make it easily comprehensible to you......

Data Handling

The word data means information in the form of numerical figures or a set of given facts. E.g. The percentage of marks scored by 10 students.......

The Triangles and its Properties

A closed figure formed by joining three non-collinear points is called a triangle. The three sides and three angles of a triangle are collectively known as elements of the triangle......

Visualising Solid Shapes

A solid is any enclosed three-dimensional shape, i.e., it has 3 dimensions- length, width and height, whereas there are some common (flat) shapes which can be easily drawn on paper. They have only.....

Fractions

Fractions having the same denominator are called like fractions, whereas fractions having different denominator are called unlike fractions......

Perimeter and Area

Mensuration : The process, art or the act of measuring is called mensuration. Anything that can be measured is said to be mensurable.......

Join ALLEN!

(Session 2025 - 26)


Choose class
Choose your goal
Preferred Mode
Choose State
  • About
    • About us
    • Blog
    • News
    • MyExam EduBlogs
    • Privacy policy
    • Public notice
    • Careers
    • Dhoni Inspires NEET Aspirants
    • Dhoni Inspires JEE Aspirants
  • Help & Support
    • Refund policy
    • Transfer policy
    • Terms & Conditions
    • Contact us
  • Popular goals
    • NEET Coaching
    • JEE Coaching
    • 6th to 10th
  • Courses
    • Online Courses
    • Distance Learning
    • Online Test Series
    • International Olympiads Online Course
    • NEET Test Series
    • JEE Test Series
    • JEE Main Test Series
  • Centers
    • Kota
    • Bangalore
    • Indore
    • Delhi
    • More centres
  • Exam information
    • JEE Main
    • JEE Advanced
    • NEET UG
    • CBSE
    • NCERT Solutions
    • NEET Mock Test
    • Olympiad
    • NEET 2025 Answer Key

ALLEN Career Institute Pvt. Ltd. © All Rights Reserved.

ISO