**Notes of chapter: Algebraic Expressions are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 8.**

**Algebraic expression –**

An **algebraic expression** is a mathematical expression that consists of terms [variables, constants and operations], exponentiation (raise to the power) and finding of roots in any combination.

**E.g.-** **(i)**3x + 5

It is an expression formed of

Variable x ,

Constant 5 and

Operation +

It is an expression formed of

Variable x ,

Constants 5 and

Operation +

Exponentiation x^{2}

**(1)Terms of an expression-**

**(a)**It can be fixed value or a **constant.**

**Eg-** 5, 4, -10

**(b)**It can be a** variable**.

**Eg:-** x, y, z

**(c)**It can be **product of two or more variables**.

**Eg:-** x^{2}, xy, xyz

**(d)**It can be **product of constant and variable**.

**Eg:**– 5x, 6xy,2xy^{2},-3x

Hence, such parts of an expression which are formed separately are known as **terms.**

The expression which is formed with the use of operations (addition, subtraction, multiplication and division) on the variables and constants is called **algebraic expression.**

**(2)Types of terms**

**(i) Like terms-**

When terms have same variable factors with same powers, they are called **like terms**.

**Eg:**– 2xy and 3xy are like terms.

**(ii) Unlike terms-**

When terms have different variable factors with different powers, they are called **unlike terms**.

**Eg:-** 2x and 2xy are unlike terms.

**(3) Types of algebraic expressions-**

**(i)Monomial-**

An expression with only one term is called a **monomial**.

**Eg:-** 2xy,-3, 2y^{2}

**(ii)Binomial-**

An expression which contains two unlike term is called a** binomial**.

**Eg:-** 2xy – 3

**(iii)Trinomial –**

An expression which contains three unlike term is called a **trinomial.**

**Eg:-** 2xy – 3 + 2y^{2}

**(4)Addition and Subtraction of like terms-**

The sum or difference of two or more like terms is a like term with a numerical coefficient equal to the sum or difference of the numerical coefficients of all the like terms.

**Eg:-**

**(i)Add** 5x and 2x

**Ans- **

5x + 2x

=(5+2)x[ distributive law]

=7x

**(ii) Subtract** 2x from5x.

**Ans-**

5x – 2x

= (5-2)x [distributive law]

=3x

**(5)Addition and Subtraction of unlike terms-**

Unlike terms can not be added or subtracted .So we write them as they are.

**Eg:-**

**(i)Add** 2x and 4y

**Ans-**

2x + 4y

**(ii)Subtract **3x from 4y

**Ans-**

4y – 3x

**(8)** **Multiplication of algebraic expressions-**

**(i) Steps for multiplication of monomial expressions-**

**Eg:- **Multiply 2x and 3y

**Step 1-**

Multiply coefficients of monomial expressions.

Coefficients of product = Coefficient of first monomial × Coefficient of second monomial

2 × 3 = 6

**Step 2-**

Multiply algebraic factors of monomial expressions.

Algebraic factor of products

= algebraic factor of first monomial × algebraic factor of second monomial

Multiplication of x and y = xy

2x × 3y = 6xy

**Some more examples-**

**(a) Multiplying two monomials-**

**(A)** 3x × 4y

= 3 × 4xy

= 12xy

**(B)** x × 4y = 4xy

**(C)** 6x × (- 2y)

= 6 × (-2) × x × y

= -12xy

**(D)** 2x × 3x^{2}

= 2 × 3 × x × x^{2}

= 6 x^{3}

**(E)**2x × (-2xyz)

= (2 × -2) × (x × xyz)

= -4x^{2}yz

**(b)** **Multiplying three or more monomials-**

**(A)** 2x × 3y × 4z

= (2 × 3 × 4) × (x × y × z)

= 24xyz

**(B)**4xy × 2x^{2}y^{2} × 4x^{3}y^{3}

= (4 × 2 × 4) × (x^{2} × x^{3}) × (y^{2} × y^{3})

= 32x^{5}y^{5}

**(ii)Multiplying a monomial by a polynomial-**

**(a) Multiplying a monomial by a binomial-**

2x × (3y + 1)

= (2x × 3y) + (2x × 1) [Distributive Law]

= 6xy + 2x

**(b) Multiplying a monomial by a trinomial-**

2x × (x + 4xy – 2)

= (2x × x) + (2x × 4xy) – (2x × 2) [Distributive Law]

= 2x^{2} + 8x^{2}y – 4x

**(iii)** **Multiplying a polynomial by a polynomial-**

**(a) Multiplying a binomial by a binomial-**

(x + y)(2x – 4y)

=x × (2x – 4y) + y × (2x – 4y)[Distributive law]

=(x × 2x) – (x × 4y) + (y × 2x) – (y × 4y)

= 2x^{2} – 4xy + 2xy – 4y^{2}

= 2x^{2} – 2xy – 4y^{2}

**(b) Multiplying a binomial by a trinomial-**

(2x + y)(x + y – 2)

=2x × (x + y – 2) + y (x + y – 2)[Distributive Law]

=(2x × x)+(2x × y) -(2x × 2) + (y × x) + ( y × y) –(y × 2)

=2x^{2}+ 2xy -4x +xy + y^{2}– 2y

=2x^{2}+ 3xy -4x + y^{2}– 2y

**(9) Identity-**

An **identity** is an equality which is true for every value of the variable in it.

**Eg:- **

(x + 1) (x + 2) = x^{2} + 3x + 2

is an identity because every value of variable(x) maintains equality.

If x = 2

Then,

(x + 1) (x + 2) = x^{2} + 3x + 2

(2 + 1)(2 + 2) = 2^{2} + 3 × 2 + 2

3 × 4 = 4 + 6 +2

12 = 12

LHS = RHS

If x = 5

Then,

(x + 1) (x + 2) = x^{2} + 3x + 2

(5+ 1)(5 + 2) = 5^{2} + 3 × 5 + 2

6 × 7 = 25 + 15 +2

42 = 42

LHS = RHS

Hence, equality is true for every value of variable.

**(10)** An** equation** is true for certain values of variable in it.

**Eg:-**

x^{2} + 3x + 2 = 12

If x = 2

LHS

=x^{2} + 3x + 2

= 2^{2} + 3 × 2 + 2

= 4 + 6 +2

= 12

=RHS

If x = 5

LHS

=x^{2} + 3x + 2

= 5^{2} + 3 × 5 + 2

= 25 + 15 +2

≠ 42

RHS

Hence, equation is true only for x = 2.

**(11)** **Standard Identities-**

**(i)** (a + b)^{2} = a^{2} + 2ab + b^{2}

**(ii)** (a – b)^{2} = a^{2} – 2ab + b^{2}

**(iii)**(a + b) (a – b) = a^{2} – b^{2}

**(iv)** (x + a) (x + b) = x^{2} + (a + b)x + ab

**(a)** These four standard identities are useful in carrying out squares and products of algebraic expressions.

**(b)** These are also useful as alternative methods to calculate products of numbers and so on.

**Helping Topics**