Algebraic Expressions and Identities Class 8 Notes is specially designed to help students understand important chapter concepts clearly and study productively. These notes will also enable students to have an effective math practice session and be ready to tackle chapter questions that could be asked in the exams.
Introduction to Algebraic Expressions and Identities
Algebraic expressions are expressions made up of variables and constants along with mathematical operators. Algebraic expressions have no sides or equal to sign like algebraic equations.
Examples of algebraic expressions are : 2x+4, 7y−3+6x, 3t2+4t−1.
Terms are the individual building blocks of expressions. They add up to form expressions. A term is a product of its factors.
For example, the expression 5xy – 3, is made up of two terms, 5xy and (-3).
Factors are those variables or constants, whose product form a term of an expression.
For example, 8, p and q are the factors of the term 8pq.
Factors are such that they can not be factorised further.
The product of factors forms a term and the summation of the terms forms an expression.
The numerical factor of a term is called the coefficient of that term.
For the terms, 6y and 2xy, the coefficient of 6y is 6 and the coefficient of 2xy is 2.
Like terms are those terms which have same variables raised to the same power. Like terms have same algebraic factors. The numerical coefficient of like terms can be different.
For example, 3x2y and 5x2y are like terms.
An expression with only one term is called a monomial.
Examples of monomials: 6x, 7pq, x2y, 9xyz , 4bc etc.
An expression which contains two unlike terms is called a binomial.
Examples of binomials: 4y−3z, x6−2, pq+1, etc.
Expressions that have more than two terms with non-zero coefficients and variables having non-negative integral exponents are called polynomials.
Examples: a+b+c+2, 7xy−8x+2+3y, 5t3−7t+k+3.
Addition and Subtraction of Algebraic Expressions
When we are adding or subtracting two algebraic expressions, we can only add or subtract like terms. The sum of two or more like terms is a like term, with a numerical coefficient equal to the sum of the numerical coefficient of all the like terms.
Similarly, the difference between two like terms is a like term with a numerical coefficient equal to the difference between the numerical coefficients of the two like terms.
Suppose if we have to add 3x2y+y+z and 4x2y+7a+5z, we will combine all the like terms and then add their numerical coefficients.
(3x2y + 4x2y) + (y) + (7a) + (z + 5z) = 7x2y + y + 7a + 6z
Multiplication of Algebraic Expressions
Multiplication of Monomials
When we multiply two monomials:
- the numerical coefficient of the terms is equal to the product of the numerical coefficient of both the terms.
- the exponent or power of each algebraic factor is equal to the sum of the exponents of that algebraic factor in both the monomials.
Multiplying two monomials:
- x×3y = x×3×y = 3×x×y = 3xy
- 3x×2y = 3×x×2×y = 3×2×x×y = 6xy
- 5x×(−2z) = 5×(−2)×x×z = −10xz
Multiplying three or more monomials:
- 2x × 3y × 5z = (2x×3y) × 5z = 6xy × 5z = 30xyz
- 4xy × 5x2y2 × 6x3y3 = (4xy×5x2y2) × 6x3y3 = 20x3y3 × 6x3y3 = 120x6y6
Distributive Property of Multiplication
The distributive property is an algebraic property that is used to multiply a single value and two or more values within a set of parenthesis.
Consider the expression : 6 × (2+4x)
=(6×2) + (6×4x)
=12 + 24x
Here, we have used distributive law to mutiply a monomial and a binomial.
Multiplication of any Polynomial
When we multiply any two polynomials, we multiply all the terms or monomials of one polynomial with all the terms of another polynomial.
When we multiply two binomials, every term in one binomial multiplies every term in the other binomial.
- Multiplying a binomial by a binomial
(3a + 4b) × (2a + 3b)
= 3a × (2a + 3b) + 4b × (2a + 3b)
= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b)
= 6a2 + 9ab + 8ab + 12b2
= 6a2 + 17ab + 12b2
When we multiply a binomial by a trinomial, each of the three terms of the trinomial is multiplied by each of the two terms of the binomial.
- Multiplying a binomial by a trinomial
(p + 4) × (p2 + 2p + 3)
= p × (p2 + 2p + 3) + 4 × (p2 + 2p + 3)
= (p3 + 2p2 + 3p) + (4p2 + 8p + 12)
= p3 + 6p2 + 11p + 12