Title Notes_Algebraic expressions and Factorization_Algebraic expressions and Factorization.pdf ✅

www.thinkiit.in Page 1 thinkIIT ALGEBRAIC EXPRESSIONS & FACTORIZATION  Constant : A symbol having a fixed numerical value is called a constant.  Variable : A symbol which takes various numerical values is called a variabale. Ex.1 We know that the perimeter P of a square of side s is given by P = 4 × s. Here, 4 is a constant a n d P and s are variables. Ex.2 The perimeter P of a rectangle of sides l and b is given by P = 2 (l + b). Here, 2 is a constant and l and b are variables.  Algebraic Expressions : A combination of constants and variables connected by the signs of fundamental operation of addition, subtraction, multiplication and division is called an algebraic expression.  Terms : Various parts of an algebraic expression which are separated by the signs of + or – are called the ‘terms’ of the expression. Ex.3 2x2 – 3xy + 5y2 is an algebraic expression consisting of three terms, namely, 2x2 ,–3xy and 5y2 . Ex.4 The expression 2x3 – 3x2 + 4x – 7 is an algebraic expression consisting of four terms, namely, 2x3 , –3x2 , 4x and – 7.  Monomial : An algebraic expression containing only one term is called a monomial. Ex.5 – 5,3y,7xy, 3 2 x 2yz, 3 5 a 2bc3 etc. are all monomials.  Binomial : An algebraic expression containing two terms is called a binomial. Ex.6 The expression 2x – 3, 3x + 2y, xyz –5 etc. are all binomials.  Trinomial : An algebraic expression containing three terms is called a trinomial. Ex.7 The expressions a – b + 2x2 + y2 – xy, x 3 – 2y3 – 3x2y 2z etc. are trinomial. Factors : Each terms in an algebraic expression is a product of one or more numbers (s) and /or literal (s). These number(s) and liteal(s) are known as the factors of that terms. A constant factor is called a numerical factor, while a variable factor is known as a literal factor.  Coefficient : In a term of an algebraic expression any of the factors with the sign of the term is called the coefficient of the other factors. Ex.8 In – 5xy, the coefficient of x is – 5y; the coefficient of y is –5x and the coefficient of xy is –5. Ex.9 In –x, the coefficient of x is –1.  Constant Term : A term of the expression having no literal factor is called a constant term.
www.thinkiit.in Page 2 thinkIIT Ex.11 In the algebraic expression x2 – xy + yz – 4, the constant term is –4.  Like and Unlike Terms : The terms having the same literal factors are called like or similar terms, otherwise they are called unlike terms. Ex.12 In the algebraic exspression 2a2b + 3ab2 – 7ab –4ba2 , we have 2 a2b and –4ba2 as like terms, whereas 3ab2 and –7ab are unlike terms.  EXAMPLES  Ex.13 Add : 7x2 – 4x + 5, – 3x2 + 2x – 1 and 5x2 –x + 9. Sol. We have, Required sum = (7x2 – 4x +5) + (–3x2 + 2x–1) + (5x2 – x+9) = 7x2 – 3x2 + 5x2 – 4x + 2x – x + 5– 1+9 [Collecting like terms] = (7 – 3 + 5)x2 + (–4 + 2 – 1)x +(5 –1+ 9) [Adding like terms] = 9x2 – 3x + 13 Ex.14 Add : 5x2 – 3 1 x + 2 5 , – 2 1 x 2 + 2 1 x – 3 1 and – 2x2 + 5 1 x – 6 1 . Sol. Required sum =        2 5 x 3 1 5x – 2 +        3 1 x – 2 1 x 2 1 – 2 +        6 1 x – 5 1 2x 2 = 5x2 – 2 1 x 2 – 2x2 – 3 1 x + 2 1 x + 5 1 x + 2 5 – 3 1 – 6 1 [Collecting like terms] =       – 2 2 1 5 – x 2 +         5 1 2 1 3 1 – x +       6 1 – 3 1 – 2 5 [Adding like term] =       2 10 –1– 4 x 2 +          30 10 15 6 x +       6 15 – 2 –1 = 2 5 x2 + 30 11 x + 2 (i) The product of two factors with like signs is positive and the product of two factors with unlike signs is negative i.e., (a) (+) × (+) = + (b) (+) × (–) = – (c) (–) × (+) = – and, (d) (–) × (–) = + (ii) If a is any variable and m, n are positive integers, then am × an = am+n For example , a3 × a5 = a3+5 = a8 , y4 × y = y4+1 = y5 etc.
www.thinkiit.in Page 3 thinkIIT Ex.15 Find the product of the following pairs of polynomials : (i) 4, 7x (ii) – 4a, 7a (iii) – 4x,7xy (iv) 4x3 , – 3xy (v) 4x, 0 Sol. We have, (i) 4 × 7x = (4 × 7) × x = 28 × x = 28 x (ii) (–4a) × (7a) = (–4 × 7) × (a×a) = –28a2 (iii) (–4x) × (7xy) = (–4 × 7) × (x × xy) = –28x1+1y = –28x2y (iv) (4x3 ) × (–3xy) = (4×–3) × (x3 × xy) = – 12 (x3+1y) = –12x4y (v) 4x × 0 = (4 × 0) × x = 0 × x = 0 Ex.16 Find the areas of rectangles with the following pairs of monomials as their length and breadth respectively : (i) (x, y) (ii) (10x, 5y) (iii) (2x2 , 5y2 ) (iv) (4a, 3a2 ) (v) (3mn, 4np) Sol. We know that the area of a rectangle is the product of its length and breadth. Length Breadth Length × Breadth = Area (i) x y x × y = xy (ii) 10 x 5y 10x × 5y = 50xy (iii) 2x2 5y2 2x2 × 5y2 = (2 × 5) × (x2 × y2 ) = 10x2y 2 (iv) 4a 3a2 4a × 3a2 = (4 × 3) × (a × a2 ) = 12 a3 (v) 3mn 4np 3mn × 4np= (3 × 4) × (m × n × n × p) = 12 mn2p Ex.17 Multiply : (i) 3ab2c 3 by 5a3b 2c (ii) 4x2yz by – 2 3 x 2yz2 (iii) – 5 8 x 2yz3 by – 4 3 xy2z (iv) 14 3 x 2y by x y 2 7 4 (v) 2.1a2bc by 4ab2 Sol. (i) We have, (3ab2c 3 ) × (5a3b 2c) = (3 × 5) × (a × a3 × b2 × b2 × c3 × c) = 15a1+3b 2+2c 3+1 = 15a4b 4c 4 (ii) We have, (4x2yz) ×       2 2 x yz 2 3 –