Algebraic Expressions and Identities Notes Class 8
In this page we will explain the topics for the chapter 8 of Algebraic Expressions and Identities Class 8 Maths.We have given quality Algebraic Expressions and Identities Class 8 Notes along with video to explain various things so that students can benefits from it and learn maths in a fun and easy manner, Hope you like them and do not forget to like , social share
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Here are the Important points
1. This expression is made up of two terms, 11x and 1.
2. Terms are added to form algebraic expressions.
3. Terms themselves can be formed as the product of factors. The term 11x is the product of its factors 11 and x. The term 1 is made up of just one factor, i.e., 1.
4. The numerical factor of a term is called its numerical coefficient or simply coefficient. The coefficient in the term 11x is 11 and the coefficient in the constant term 1 is 1.
Monomials, Binomials, trinomial and Polynomials
Monomials
Algebraic expression having one terms is called monomials
Example
$11x^2$
$6xy$
$-2z$
$10y$
$-9$
$82mnq$
Binomials
Algebraic expression having two terms is called Binomials
Example
$x + xy$
$2p + 11q$
$y + 1$
$11 - 3xy$
Trinomial
Algebraic expression having three terms is called Trinomial
In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial
Example
$x + y + z + 1$
$3pq$
$11xyz - 10x$
$4x + 2y + z$
Watch this tutorial for more explanation about polynomials,monomials,binomials
Like and Unlike Terms
It is quite important concept and quite well used in addition, subtraction and multiplication of the algebraic expression.
When the variable part of the terms is same, they are called like terms
Example
$2x$, $3x$ are like term
$5x^2$ and $-9x^2$ are like terms.
Like term can be add or subtracted. Basically We need to add or subtract the coefficient
When the variable part of the terms is not same, they are called unlike terms
Example
$2x$, $3y$ are unlike term
$5x^2$ and $-9zx^2$ are unlike terms.
Addition and Subtraction of Algebraic Expressions
Addition
We many times need to add the two algebraic expression. Adding algebraic expression just means adding the like terms. We need to follow below steps for Addition of algebraic expression
1. We write each expression to be added in a separate row. While doing so we write like terms one below the other
Or
We add the expression together on the same line and arrange the like term together
2. Add the like terms
3. Write the Final algebraic expression
Example
Add the following expression $x - y + xy$, $y - z + yz$, $z - x + zx$ Solution
$= (x - y + xy)+( y - z + yz)+( z - x + zx)$
Arranging the like term together
$= x-x -y+y-z+z+xy+yz+zx$
$=xy+yz+zx$ as $x-x=y-y=z-z=0$
Subtraction
Similarly, for subtracting algebraic expression
1. We write each expression to be subtracted in a separate row. While doing so we write like terms one below the other and then we change the sign of the expression which is to be subtracted i.e. + becomes - and - becomes +
Or
We subtract the expression together on the same line, change the sign of all the term which is to be subtracted and then arrange the like term together
2. Add the like terms
3. Write the Final algebraic expression
Example
Subtract $4x - 7xy + 3y + 12$ from $12x - 9xy + 5y - 3$ Solution
$=(12x - 9xy + 5y - 3) - (4x - 7xy + 3y + 12)$ While subtracting, we need to remember signs are reversed after -sign once bracket is opened ie. + becomes - and - becomes +
$=12x-9xy+5y-3 -4x+7xy-3y-12$
Arranging the like term together
$=8x-2xy+2y-15$
Watch this tutorial for questions on addition and subtraction
Multiplication of Algebraic expression
General steps for Multiplication
We have to use distributive law and distribute each term of the first polynomial to every term of the second polynomial.
when you multiply two terms together you must multiply the coefficient (numbers) and add the exponents
Also as we already know ++ equals =, +- or -+ equals - and -- equals +
group like terms
Multiplication may involve
Multiplication of monomial to monomial
Multiplication of monomial to binomial, trinomial or more terms polynomials
Multiplication of binomial, trinomial or more terms polynomials to monomial
Multiplication of binomial to binomial, trinomial or more terms polynomials
Multiplication of trinomial to trinomial or more terms polynomials
Multiple the Monomials
(i) $a^2 \times (2b^{22}) \times (4a^{26})$
(ii $(\frac {-10pq^3}{3} \times (\frac {6p^3}{5})$ Answer:
We will use the below property extensively in above questions
$x^m \times x^n \times x^o=x^{m+n+o}$
1. Multiplying the constant and using the exponent property given above
we get
$a^2 \times (2b^{22}) \times (4a^{26})=8a^{28}b^{22}$
2. $(\frac {-10pq^3}{3} \times (\frac {6p^3}{5})$
$=(-12p^4q^3)$
Multiply the binomials.
(i) $(2a + 6b)$ and $(4a - 3b)$
(ii) $(x - 1)$ and $(3x - 2)$ Answer:
Let first understand how to multiply the terms $(a+b)$ and $(c+d)$
Multiplication can be done by distributive law
$( a+b) (c+d)= a(c+d) + b( c+d)$
$=(a \times c)+(a \times d)+(b \times c)+(b \times d)$
We will use the same concept in all the question below
(i) (2a + 6b) and (4a - 3b)
$=2a \times 4a - 2a \times 3b + 6b \times 4a - 6b \times 3b$
= 8a² - 6ab + 24ab -18b2
= 8a² + 18ab -18b2
(ii) (x - 1) and (3x - 2)
= x × 3x - 2x - 1 × 3x + 2
= 3x2 -2x - 3x + 2
= 3x2 - 5x + 2
Multiply Binomial by Trinomial
(3x + 1)(4x2 - 7x + 1) Answer By distributive law
=3x(4x2 - 7x + 1) + 1(4x2 - 7x + 1)
=12x3 -21x2 +3x +4x2 -7x+1
=12x3 -17x2 -4x+1 Multiply Trinomial by Trinomial
$(a + b + c)(a + b - c)$ Answer By distributive law
$=a(a + b -c) + b(a + b -c)+c(a + b -c)$
$=a^2 +ab-ac +ab +b^2 -bc +ac+bc-c^2$
$=a^2 + b^2 -c^2 +2ab$ Practice Questions
Watch this tutorial for questions on multiplication of monomials with monomials
Watch this tutorial for questions on multiplication of binomials with binomials
What is an Identity
An identity is an equality, which is true for all values of the variables in the equality.
$(a + b)^2 = a^2 + 2ab + b^2$
It is true for all the values of a and b
On the other hand, an equation is true only for certain values of its variables. An equation is not an identity
$x^2=1$
The below four identities are useful in carrying out squares and products of algebraic expressions.
$(a + b)^2 = a^2 + 2ab + b^2$
$(a - b)^2 = a^2 - 2ab + b^2$
$(a + b) (a - b) = a^2- b^2$
$(x + a) (x + b) = x^2 + (a + b) x + ab$
These allow easy alternative methods to calculate products of numbers.
Example
Use a suitable identity to get each of the following products.
(i) $(y + 1) (y + 1)$
(ii) $(2x + 1) (2x -1)$
(iii) $(2z - 3) (2z - 3)$ Solution
(i) $(y + 1) (y + 1)$
$=(y+1)^2$
Now $(a + b)^2 = a^2 + 2ab + b^2$
$=y^2 +2y+1$
(ii) $(2x + 1) (2x -1)$
Now $(a + b) (a - b) = a^2- b^2$
$=4x^2 -1$
(iii) $(2z - 3) (2z - 3)$
$=(2z-3)^2$
Now as $(a - b)^2 = a^2 - 2ab + b^2$
$=4z^2 -12z+9$ Practice Question
$(x-1)(x-1)$
$(y+3)(y+3)$
$(2z+1)(2z-1)$
$(x-9)(x+10)$
$(x+7)(x+7)$
$(x-5)(x-7)$
Watch this tutorial for questions on identities
Frequently asked Questions on CBSE Class 8 Maths Chapter 9: Algebraic Expressions and Identities
What is the difference between equation and identity?
An identity is an equality, which is true for all values of the variables in the equality.
On the other hand, an equation is true only for certain values of its variables. An equation is not an identity.
What are the 4 identities of algebraic expressions Class 8?
$(a + b)^2 = a^2 + 2ab + b^2$
$(a - b)^2 = a^2 - 2ab + b^2$
$(a + b) (a - b) = a^2- b^2$
$(x + a) (x + b) = x^2 + (a + b) x + ab$
Can an algebraic expression have more than one term?
An algebraic expression can have one or more terms example 1 , 1+x
What are basic algebraic expressions?
A Basic Algebraic expression is the expression having constants and 1 variable that is connected to each other by addition ,subtrcation or multiplication example 5x, 5+x, x-5
What are Types of algebraic expressions?
An Algebraic expression can be monomial, bonomial, trinomial or polynomial
Summary
Here is Algebraic Expressions and Identities Class 8 Notes Summary
![Algebraic Expressions and Identities Class 8 Notes](multiplication-of-binomial-by-binomial.PNG")
