Class 8 Maths Chapter 9 Notes - Algebric Expressions And Identities

Introduction to Algebraic expressions and identities

An algebraic expression is made up of constants, variables, and mathematical operations (addition, subtraction, multiplication and/or division). A constant is number whose value does not change, whereas the value of a variable changes. For example, x + 9 is an algebraic expression where 9 is a constant and x is a variable. Different elements like terms, factors and coefficients form these expressions.

Each part of an algebraic expression separated by a plus or minus sign is called a term of the algebraic expression.

Factors are the numbers and/or variables whose product make a term. If we look at the term 7x, it is the product of two factors 7 and x.

Coefficient is defined as the number that is multiplied by the other variables. In the term 11pqr, 11 is the coefficient of pqr.

An expression can be termed as monomial, binomial, trinomial, and polynomial based on the number of terms it has.

Monomial: It is an algebraic expression with only one term. For example, 9y is a monomial.

Binomial: It is an algebraic expression with two terms. For example, 2a – 6b is a binomial.

Trinomial: It is an algebraic expression with three terms. For example, 9a + 5b – c .

Polynomial: It is an algebraic expression that contains one or more terms consisting of constants and/or variables (whose exponents are non-negative integers). Monomials, binomials and trinomials come under the category of polynomials.

The terms of an algebraic expression can be categorised as like terms and unlike terms.

Like and Unlike Terms

Like Terms: If two or more terms have the same variable, then they are called like terms.

For example, 7x + 8x is an algebraic expression with like terms.

Unlike Terms: If two or more terms have different variables, they are called unlike terms.

For example, 7x + 8y is an algebraic expression with unlike terms.

We should understand how to select like and unlike terms because this helps us to add or subtract the terms of an algebraic expression. Let us see how we can do this.

NCERT Solutions for Class 8 Maths Chapter 9
NCERT Solutions Class 8 Maths Chapter 9
NCERT Solutions Class 8 Maths Exercise 9.1
NCERT Solutions Class 8 Maths Exercise 9.2
NCERT Solutions Class 8 Maths Exercise 9.3
NCERT Solutions Class 8 Maths Exercise 9.4
NCERT Solutions Class 8 Maths Exercise 9.5

Addition and Subtraction

While adding and subtracting two or more algebraic expressions, we first group the like terms. Then we add or subtract them using simple number rules.

If a and b are coefficients and x and y are variables, ax + bx + y = (a + b)x + y.

If a and b are coefficients and x and y are variables, ax − bx + y = (a − b)x + y.

✍ Also learn: Addition and subtraction of algebraic expressions class 8

Let us see an example.

Subtract x² + 3y² + 4xy – 4xyz from 9x² – 4y² + 7y + 2xy + 6.

9x² – 4y² + 7y + 2xy + 6

x² + 3y² + 4xy – 4xyz

– – – +

---------------------------------------------------

8x² – 7y² + 7y − 2xy + 6 + 4xyz

Multiplication

While multiplying algebraic expressions, every term of the first expression is multiplied with every term of the second expression. We have to follow some other steps as well.

Multiplying Like Terms

● The coefficients of the terms will get multiplied.

● The powers of the variables will not get multiplied, but added.

● Example: The product of 4x² and 9x will be 36x³.

Multiplying Unlike Terms

● The coefficients of the terms will get multiplied.

● The power will remain the same if the variables are different.

● If some of the variables are the same, then their powers will be added.

● Example: The product of xy, 4x, and 9xz will be 36x³yz.

Let’s see the multiplication of different types of algebraic expressions

Monomial by Monomial: If a and b are coefficients and x and y are variables, ax × bxy = abx²y.

Monomial by Binomial: If a and b are coefficients and x and y are variables, ax × (x + bxy) = ax²+ abx²y.

Monomial by Trinomial: If a and b are coefficients and x, y, and z are variables, ax × (x + by – z) = ax²+ abxy – axz.

Binomial by Binomial: If a and b are coefficients and x, y, and z are variables, (ax + y) × (x – z) = ax²– axz + xy − yz.

Binomial by Trinomial: If a and b are coefficients and x, y, and z are variables, (ax + y) × (x + by – z) = ax²+ abxy – axz.

Let us see an example.

Find the value of (p + 2q + r) × (2p – 3q).

(p + 2q + r) × (2p – 3q) = p (2p – 3q) + 2q (2p – 3q) + r (2p – 3q)

(p + 2q + r) × (2p – 3q) = p × 2p – p × 3q + 2q × 2p – 3q × 2q + r × 2p – r × 3q

(p + 2q + r) × (2p – 3q) = 2p² – 3pq + 4pq – 6q² + 2pr – 3qr

(p + 2q + r) × (2p – 3q) = 2p² + pq – 6q² + 2pr – 3qr

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Identities

Equations contain an ‘equal to’ operator along with some other expressions. In an equation, if for every value of the variable, the value of the expression on the LHS is equal to the value of the expression on the RHS, then the equation is an identity. We will learn four standard identities that we generally use.

● (a + b)² = a² + 2ab + b²

● (a – b)² = a² – 2ab + b²

● (a + b)(a – b) = a² – b²

● (x + a)(x + b) = x² + (a + b)x + ab

Let us see an example where identities are used.

Find the value of (4x² + 4xy) (4x² + 3xy).

Using the identity: (x + a) (x + b) = x² + (a + b) x + ab
(4x² + 4xy) (4x² + 3xy)

= (4x²)² + (4xy + 3xy)(4x²) + 4xy × 3xy

= 16x⁴ + (16x³y + 12x³y) + 12x²y²

= 16x⁴ + 28x³y + 12x²y²

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