Algebraic Expressions and Identities Class 8 Notes CBSE Maths Chapter 9 (Free PDF Download)

What are Expressions?

• Expressions are mathematical sentences which include at least two constant, variables or both connected by mathematical operations.

• For example: \[3+5,x-y,5+y\] etc.

• Number line and expression: An expression can be shown on a number line as shown by the example below

• For example: To show $x+3$ on a number line 

• Make $x$ at any distance on the number line then move $3$ units right of $x$. The point P shows $x+3$ on a number line 

• Similarly, if we have to show $x-4$ on the number line Make $x$ at any distance on the number line then move $4$ units left of $x$.

Terms, Factors and Coefficients

• Terms are the building blocks of an expression. It may be single number, single variable, product of variables, product of numbers or product of number and variable both.

For example: In expression $3x+5y$ there are two terms $3x$ and $5y$ 

• Factors are number or algebraic expressions that completely divide another number or expression. 

For example: In expression $5y$ the factors are $5$ and $y$ 

• The numerical factor of a term is called coefficients

For example: In expression $5x+7y$ , $5$ and $7$ are the coefficients of $x$ and $y$  respectively

Monomials, Binomials, Polynomials

• Expression that contains only one term is called monomial 

For example: $5y,3x,{{x}^{2}},9,ab$ etc. are monomials 

• Expression that contains two term is called binomial 

For example: $2x+y,4{{y}^{2}}+z,ab+mn$ etc. are binomials

• Expression that contains three term is called trinomial 

For example: $x+y+z,2a+b+7c,{{a}^{2}}+6b+{{c}^{2}}$ etc.

• An expression containing one or more terms with non-zero coefficient and with variables having non negative exponents is called a polynomial.

• A polynomial may contain any number of terms.

For example: $2x+y,4xy,a+b+c,{{m}^{2}}$ etc. are polynomials

Like and Unlike Terms

• Like terms are those terms which have the same variable, power of the variable should also be same and coefficient can be different.

For example: $6x$ and $8x$ are the like terms 

• Unlike terms are opposite of like terms it has different variables.

For example: $5y$ and $9a$ are unlike terms

Addition and Subtraction of Algebraic Expressions

• In addition, like terms are added.

For example: Add of $3a+5b+2ab$ and $2a+3b+7ab$ 

Write like terms below one another then add 

$3a+5b+2ab $ 

$+2a+3b+7ab $

$ \overline{5a+8b+9ab} $ 

• In subtraction also only like terms are subtracted.

For example: Subtract ${{x}^{2}}+2x-4xy$ and $3{{x}^{2}}+2xy$ 

Write like terms below one another then subtract 

$ {{x}^{2}}+2x-4xy $ 

$ +3{{x}^{2}}+2xy $

$ \overline{-2{{x}^{2}}+2x-6xy} $ 

Multiplication of Algebraic Expression

1. Multiplying a Monomial by a Monomial

• The product of two monomial is always a monomial

• For example: Multiplication of $20xy$ and $4z$  is $20xy\times 4z=80xyz$ 

Multiplication of $7z,2{{x}^{2}}$ and $4y$ is $7z\times 2{{x}^{2}}\times 4y=56{{x}^{2}}yz$ 

2. Multiplying a Monomial by a Polynomial

• Use distributive law to multiply a monomial by a polynomial i.e., multiply monomial to each term of polynomial

• For example: Multiplication of $6x$ and $3{{x}^{2}}+5y$ is 

$\left( 6x \right)\times \left( 3{{x}^{2}}+5y \right)=\left( 6x\times 3{{x}^{2}} \right)+\left( 6x\times 5y \right)$ 

$=18{{x}^{3}}+30xy$ 

3. Multiplying a Polynomial by a Polynomial

• Multiply each term of a polynomial by each term of another polynomial then combine like terms if any

• For example: Multiplication of $\left( 2x+3y \right)$ and $\left( 2x-3y+z \right)$ is

$\left( 2x+3y \right)\times \left( 2x-3y+z \right)=\left( 2x \right)\left( 2x-3y+z \right)+\left( 3y \right)\left( 2x-3y+z \right)$ 

$=\left( \left( 2x \right)\left( 2x \right)-\left( 2x \right)\left( 3y \right)+\left( 2x \right)\left( z \right) \right)+\left( \left( 3y \right)\left( 2x \right)-\left( 3y \right)\left( 3y \right)+\left( 3y \right)\left( z \right) \right)$ 

$=\left( 4{{x}^{2}}-6xy+2xz \right)+\left( 6xy-9{{y}^{2}}+3yz \right)$ 

$=4{{x}^{2}}+2xz-9{{y}^{2}}+3yz$ 

Identity

• An equation is not true for any value of variable but for certain values.

• The equation which is always true for any value of variables is called identity.

• Standard Identities

The following identities are known as standard identities

1. ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ 

2. ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ 

3. $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ 

• Another Useful Identity

$\left( x+a \right)\left( x+b \right)={{x}^{2}}+\left( a+b \right)x+ab$ 

• These identities make calculation easier.

Download Revision Notes Class 8 Maths Chapter 9 - Free PDF

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A Few Glimpses of Chapter 9 Algebraic Expressions and Identities

With the previous basic knowledge of algebraic expression or simple expressions such as 9y, 5p - 4q, 3p² -q,  etc, Class 8 Chapter 9 Algebraic Expressions and Identities will help you to explore the world of Algebraic Expression.

Algebraic Expressions

An algebraic expression is an expression that is formed using variables and constants. The expression 9y - 5 is formed using variable y and constant 9 and 5. We know that the value of y in expression 9y - 5 can be anything. It can be either 9, 5, 0, 9/5. etc. There can be unlimited different values. It is important to note that the value of expression varies with the value chosen for the variables it includes. Hence, the value of 9y - 5 goes on changing as y takes different values. For example, when y = 2, 9(2) - 5 = 18 - 5 = 13, when y = 0 , 9(0) - 5 = - 5 etc.

Algebraic Identities

In Mathematics, algebraic identity is an equality that holds despite the value chosen for its variables. Generally, Algebraic identities are used to simplify or rearrange algebraic expression. Through definition, we can state that two sides of identities are replaceable., so we can easily replace one with the other at any time. For example, the identity ( p + q)² = p² + q² + 2pq is true for all the choices of p and q, whether they are complex or real numbers.

Chapter 9 Algebraic Expressions and Identities includes seven exercises and 9 different topics and the Class 8 Maths Revision Notes Chapter 9 PDF provided by Vedantu includes the short and brief explanation of every topic given in the chapter. Now, let us look at the topics discussed in this chapter.

• What are Expressions

• Terms, Factors, Coefficients

• Monomials, Binomials, Polynomials

• Addition and Subtraction of Algebraic Expressions

• Like and Unlike Terms

• Multiplication of  Algebraic Expressions

• Multiplication of two Monomials

• Multiplication of three or more Monomials

• Multiplication of Monomials by Binomials

• Multiplication of Monomials by Trinomials

• Multiplication of Polynomial by Polynomial

• Multiplication of Binomial by Binomial

• Multiplication of Binomial by Trinomial

• What is Identity?

• Standard Identities

• Applying Identities

Download Class 8 Maths Revision Notes Chapter 9 now and gets the brief explanations of all the topics discussed above.

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