Page 1 ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x 2 , 4xy + 7 etc. Y ou can form many more expressions. As you know expressions are formed from variables and constants. The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7. We know that, the value of y in the expression, 2y – 5, may be anything. It can be 2, 5, –3, 0, 57 ,– 23 etc.; actually countless different values. The value of an expression changes with the value chosen for the variables it contains. Thus as y takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc. Find the value of the expression 2y – 5 for the other given values of y. Number line and an expression: Consider the expression x + 5. Let us say the variable x has a position X on the number line; X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P , 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on. What about the position of 4x and 4x + 5? The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C. Algebraic Expressions and Identities CHAPTER 9 Page 2 ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x 2 , 4xy + 7 etc. Y ou can form many more expressions. As you know expressions are formed from variables and constants. The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7. We know that, the value of y in the expression, 2y – 5, may be anything. It can be 2, 5, –3, 0, 57 ,– 23 etc.; actually countless different values. The value of an expression changes with the value chosen for the variables it contains. Thus as y takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc. Find the value of the expression 2y – 5 for the other given values of y. Number line and an expression: Consider the expression x + 5. Let us say the variable x has a position X on the number line; X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P , 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on. What about the position of 4x and 4x + 5? The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C. Algebraic Expressions and Identities CHAPTER 9 138 MATHEMATICS TRY THESE TRY THESE 1. Give five examples of expressions containing one variable and five examples of expressions containing two variables. 2. Show on the number line x, x – 4, 2x + 1, 3x – 2. 9.2 Terms, Factors and Coefficients Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor, i.e., 5. The expression 7xy – 5x has two terms 7xy and –5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its numerical coefficient or simply coefficient. The coefficient in the term 7xy is 7 and the coefficient in the term –5x is –5. 9.3 Monomials, Binomials and Polynomials Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one. Examples of monomials: 4x 2 , 3xy, –7z, 5xy 2 , 10y, –9, 82mnp, etc. Examples of binomials: a + b, 4l + 5m, a + 4, 5 –3xy, z 2 – 4y 2 , etc. Examples of trinomials: a + b + c, 2x + 3y – 5, x 2 y – xy 2 + y 2 , etc. Examples of polynomials: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc. 1. Classify the following polynomials as monomials, binomials, trinomials. – z + 5, x + y + z, y + z + 100, ab – ac, 17 2. Construct (a) 3 binomials with only x as a variable; (b) 3 binomials with x and y as variables; (c) 3 monomials with x and y as variables; (d) 2 polynomials with 4 or more terms. 9.4 Like and Unlike Terms Look at the following expressions: 7x, 14x, –13x, 5x 2 , 7y, 7xy, –9y 2 , –9x 2 , –5yx Like terms from these are: (i) 7x, 14x, –13x are like terms. (ii) 5x 2 and –9x 2 are like terms. TRY THESE Identify the coefficient of each term in the expression x 2 y 2 – 10x 2 y + 5xy 2 – 20. Page 3 ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x 2 , 4xy + 7 etc. Y ou can form many more expressions. As you know expressions are formed from variables and constants. The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7. We know that, the value of y in the expression, 2y – 5, may be anything. It can be 2, 5, –3, 0, 57 ,– 23 etc.; actually countless different values. The value of an expression changes with the value chosen for the variables it contains. Thus as y takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc. Find the value of the expression 2y – 5 for the other given values of y. Number line and an expression: Consider the expression x + 5. Let us say the variable x has a position X on the number line; X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P , 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on. What about the position of 4x and 4x + 5? The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C. Algebraic Expressions and Identities CHAPTER 9 138 MATHEMATICS TRY THESE TRY THESE 1. Give five examples of expressions containing one variable and five examples of expressions containing two variables. 2. Show on the number line x, x – 4, 2x + 1, 3x – 2. 9.2 Terms, Factors and Coefficients Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor, i.e., 5. The expression 7xy – 5x has two terms 7xy and –5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its numerical coefficient or simply coefficient. The coefficient in the term 7xy is 7 and the coefficient in the term –5x is –5. 9.3 Monomials, Binomials and Polynomials Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one. Examples of monomials: 4x 2 , 3xy, –7z, 5xy 2 , 10y, –9, 82mnp, etc. Examples of binomials: a + b, 4l + 5m, a + 4, 5 –3xy, z 2 – 4y 2 , etc. Examples of trinomials: a + b + c, 2x + 3y – 5, x 2 y – xy 2 + y 2 , etc. Examples of polynomials: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc. 1. Classify the following polynomials as monomials, binomials, trinomials. – z + 5, x + y + z, y + z + 100, ab – ac, 17 2. Construct (a) 3 binomials with only x as a variable; (b) 3 binomials with x and y as variables; (c) 3 monomials with x and y as variables; (d) 2 polynomials with 4 or more terms. 9.4 Like and Unlike Terms Look at the following expressions: 7x, 14x, –13x, 5x 2 , 7y, 7xy, –9y 2 , –9x 2 , –5yx Like terms from these are: (i) 7x, 14x, –13x are like terms. (ii) 5x 2 and –9x 2 are like terms. TRY THESE Identify the coefficient of each term in the expression x 2 y 2 – 10x 2 y + 5xy 2 – 20. ALGEBRAIC EXPRESSIONS AND IDENTITIES 139 TRY THESE (iii) 7xy and –5yx are like terms. Why are 7x and 7y not like? Why are 7x and 7xy not like? Why are 7x and 5x 2 not like? Write two terms which are like (i) 7xy (ii) 4mn 2 (iii) 2l 9.5 Addition and Subtraction of Algebraic Expressions In the earlier classes, we have also learnt how to add and subtract algebraic expressions. For example, to add 7x 2 – 4x + 5 and 9x – 10, we do 7x 2 – 4x +5 +9x –10 7x 2 + 5x –5 Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown. Thus 5 + (–10) = 5 –10 = –5. Similarly, – 4x + 9x = (– 4 + 9)x = 5x. Let us take some more examples. Example 1: Add: 7xy + 5yz – 3zx, 4yz + 9zx – 4y , –3xz + 5x – 2xy. Solution: Writing the three expressions in separate rows, with like terms one below the other, we have 7xy + 5yz –3zx +4yz + 9zx – 4y +–2xy –3zx + 5x (Note xz is same as zx) 5xy + 9yz +3zx + 5x – 4y Thus, the sum of the expressions is 5xy + 9yz + 3zx + 5x – 4y. Note how the terms, – 4y in the second expression and 5x in the third expression, are carried over as they are, since they have no like terms in the other expressions. Example 2: Subtract 5x 2 – 4y 2 + 6y – 3 from 7x 2 – 4xy + 8y 2 + 5x – 3y. Solution: 7x 2 – 4xy + 8y 2 + 5x –3y 5x 2 –4y 2 +6y – 3 (–) (+) (–) (+) 2x 2 – 4xy +12y 2 + 5x – 9y + 3 Page 4 ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x 2 , 4xy + 7 etc. Y ou can form many more expressions. As you know expressions are formed from variables and constants. The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7. We know that, the value of y in the expression, 2y – 5, may be anything. It can be 2, 5, –3, 0, 57 ,– 23 etc.; actually countless different values. The value of an expression changes with the value chosen for the variables it contains. Thus as y takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc. Find the value of the expression 2y – 5 for the other given values of y. Number line and an expression: Consider the expression x + 5. Let us say the variable x has a position X on the number line; X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P , 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on. What about the position of 4x and 4x + 5? The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C. Algebraic Expressions and Identities CHAPTER 9 138 MATHEMATICS TRY THESE TRY THESE 1. Give five examples of expressions containing one variable and five examples of expressions containing two variables. 2. Show on the number line x, x – 4, 2x + 1, 3x – 2. 9.2 Terms, Factors and Coefficients Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor, i.e., 5. The expression 7xy – 5x has two terms 7xy and –5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its numerical coefficient or simply coefficient. The coefficient in the term 7xy is 7 and the coefficient in the term –5x is –5. 9.3 Monomials, Binomials and Polynomials Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one. Examples of monomials: 4x 2 , 3xy, –7z, 5xy 2 , 10y, –9, 82mnp, etc. Examples of binomials: a + b, 4l + 5m, a + 4, 5 –3xy, z 2 – 4y 2 , etc. Examples of trinomials: a + b + c, 2x + 3y – 5, x 2 y – xy 2 + y 2 , etc. Examples of polynomials: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc. 1. Classify the following polynomials as monomials, binomials, trinomials. – z + 5, x + y + z, y + z + 100, ab – ac, 17 2. Construct (a) 3 binomials with only x as a variable; (b) 3 binomials with x and y as variables; (c) 3 monomials with x and y as variables; (d) 2 polynomials with 4 or more terms. 9.4 Like and Unlike Terms Look at the following expressions: 7x, 14x, –13x, 5x 2 , 7y, 7xy, –9y 2 , –9x 2 , –5yx Like terms from these are: (i) 7x, 14x, –13x are like terms. (ii) 5x 2 and –9x 2 are like terms. TRY THESE Identify the coefficient of each term in the expression x 2 y 2 – 10x 2 y + 5xy 2 – 20. ALGEBRAIC EXPRESSIONS AND IDENTITIES 139 TRY THESE (iii) 7xy and –5yx are like terms. Why are 7x and 7y not like? Why are 7x and 7xy not like? Why are 7x and 5x 2 not like? Write two terms which are like (i) 7xy (ii) 4mn 2 (iii) 2l 9.5 Addition and Subtraction of Algebraic Expressions In the earlier classes, we have also learnt how to add and subtract algebraic expressions. For example, to add 7x 2 – 4x + 5 and 9x – 10, we do 7x 2 – 4x +5 +9x –10 7x 2 + 5x –5 Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown. Thus 5 + (–10) = 5 –10 = –5. Similarly, – 4x + 9x = (– 4 + 9)x = 5x. Let us take some more examples. Example 1: Add: 7xy + 5yz – 3zx, 4yz + 9zx – 4y , –3xz + 5x – 2xy. Solution: Writing the three expressions in separate rows, with like terms one below the other, we have 7xy + 5yz –3zx +4yz + 9zx – 4y +–2xy –3zx + 5x (Note xz is same as zx) 5xy + 9yz +3zx + 5x – 4y Thus, the sum of the expressions is 5xy + 9yz + 3zx + 5x – 4y. Note how the terms, – 4y in the second expression and 5x in the third expression, are carried over as they are, since they have no like terms in the other expressions. Example 2: Subtract 5x 2 – 4y 2 + 6y – 3 from 7x 2 – 4xy + 8y 2 + 5x – 3y. Solution: 7x 2 – 4xy + 8y 2 + 5x –3y 5x 2 –4y 2 +6y – 3 (–) (+) (–) (+) 2x 2 – 4xy +12y 2 + 5x – 9y + 3 140 MATHEMATICS Note that subtraction of a number is the same as addition of its additive inverse. Thus subtracting –3 is the same as adding +3. Similarly , subtracting 6y is the same as adding – 6y; subtracting – 4y 2 is the same as adding 4y 2 and so on. The signs in the third row written below each term in the second row help us in knowing which operation has to be performed. EXERCISE 9.1 1. Identify the terms, their coefficients for each of the following expressions. (i) 5xyz 2 – 3zy (ii) 1 + x + x 2 (iii) 4x 2 y 2 – 4x 2 y 2 z 2 + z 2 (iv) 3 – pq + qr – rp (v) 22 xy xy +- (vi) 0.3a – 0.6ab + 0.5b 2. Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories? x + y, 1000, x + x 2 + x 3 + x 4 , 7 + y + 5x, 2y – 3y 2 , 2y – 3y 2 + 4y 3 , 5x – 4y + 3xy, 4z – 15z 2 , ab + bc + cd + da, pqr, p 2 q + pq 2 , 2p + 2q 3. Add the following. (i) ab – bc, bc – ca, ca – ab (ii) a – b + ab, b – c + bc, c – a + ac (iii) 2p 2 q 2 – 3pq + 4, 5 + 7pq – 3p 2 q 2 (iv) l 2 + m 2 , m 2 + n 2 , n 2 + l 2 , 2lm + 2mn + 2nl 4. (a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3 (b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz (c) Subtract 4p 2 q – 3pq + 5pq 2 – 8p + 7q – 10 from 18 – 3p – 11q + 5pq – 2pq 2 + 5p 2 q 9.6 Multiplication of Algebraic Expressions: Introduction (i) Look at the following patterns of dots. Pattern of dots Total number of dots 4 × 9 5 × 7 Page 5 ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x 2 , 4xy + 7 etc. Y ou can form many more expressions. As you know expressions are formed from variables and constants. The expression 2y – 5 is formed from the variable y and constants 2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7. We know that, the value of y in the expression, 2y – 5, may be anything. It can be 2, 5, –3, 0, 57 ,– 23 etc.; actually countless different values. The value of an expression changes with the value chosen for the variables it contains. Thus as y takes on different values, the value of 2y – 5 goes on changing. When y = 2, 2y – 5 = 2(2) – 5 = –1; when y = 0, 2y – 5 = 2 × 0 –5 = –5, etc. Find the value of the expression 2y – 5 for the other given values of y. Number line and an expression: Consider the expression x + 5. Let us say the variable x has a position X on the number line; X may be anywhere on the number line, but it is definite that the value of x + 5 is given by a point P , 5 units to the right of X. Similarly, the value of x – 4 will be 4 units to the left of X and so on. What about the position of 4x and 4x + 5? The position of 4x will be point C; the distance of C from the origin will be four times the distance of X from the origin. The position D of 4x + 5 will be 5 units to the right of C. Algebraic Expressions and Identities CHAPTER 9 138 MATHEMATICS TRY THESE TRY THESE 1. Give five examples of expressions containing one variable and five examples of expressions containing two variables. 2. Show on the number line x, x – 4, 2x + 1, 3x – 2. 9.2 Terms, Factors and Coefficients Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor, i.e., 5. The expression 7xy – 5x has two terms 7xy and –5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its numerical coefficient or simply coefficient. The coefficient in the term 7xy is 7 and the coefficient in the term –5x is –5. 9.3 Monomials, Binomials and Polynomials Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one. Examples of monomials: 4x 2 , 3xy, –7z, 5xy 2 , 10y, –9, 82mnp, etc. Examples of binomials: a + b, 4l + 5m, a + 4, 5 –3xy, z 2 – 4y 2 , etc. Examples of trinomials: a + b + c, 2x + 3y – 5, x 2 y – xy 2 + y 2 , etc. Examples of polynomials: a + b + c + d, 3xy, 7xyz – 10, 2x + 3y + 7z, etc. 1. Classify the following polynomials as monomials, binomials, trinomials. – z + 5, x + y + z, y + z + 100, ab – ac, 17 2. Construct (a) 3 binomials with only x as a variable; (b) 3 binomials with x and y as variables; (c) 3 monomials with x and y as variables; (d) 2 polynomials with 4 or more terms. 9.4 Like and Unlike Terms Look at the following expressions: 7x, 14x, –13x, 5x 2 , 7y, 7xy, –9y 2 , –9x 2 , –5yx Like terms from these are: (i) 7x, 14x, –13x are like terms. (ii) 5x 2 and –9x 2 are like terms. TRY THESE Identify the coefficient of each term in the expression x 2 y 2 – 10x 2 y + 5xy 2 – 20. ALGEBRAIC EXPRESSIONS AND IDENTITIES 139 TRY THESE (iii) 7xy and –5yx are like terms. Why are 7x and 7y not like? Why are 7x and 7xy not like? Why are 7x and 5x 2 not like? Write two terms which are like (i) 7xy (ii) 4mn 2 (iii) 2l 9.5 Addition and Subtraction of Algebraic Expressions In the earlier classes, we have also learnt how to add and subtract algebraic expressions. For example, to add 7x 2 – 4x + 5 and 9x – 10, we do 7x 2 – 4x +5 +9x –10 7x 2 + 5x –5 Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them, as shown. Thus 5 + (–10) = 5 –10 = –5. Similarly, – 4x + 9x = (– 4 + 9)x = 5x. Let us take some more examples. Example 1: Add: 7xy + 5yz – 3zx, 4yz + 9zx – 4y , –3xz + 5x – 2xy. Solution: Writing the three expressions in separate rows, with like terms one below the other, we have 7xy + 5yz –3zx +4yz + 9zx – 4y +–2xy –3zx + 5x (Note xz is same as zx) 5xy + 9yz +3zx + 5x – 4y Thus, the sum of the expressions is 5xy + 9yz + 3zx + 5x – 4y. Note how the terms, – 4y in the second expression and 5x in the third expression, are carried over as they are, since they have no like terms in the other expressions. Example 2: Subtract 5x 2 – 4y 2 + 6y – 3 from 7x 2 – 4xy + 8y 2 + 5x – 3y. Solution: 7x 2 – 4xy + 8y 2 + 5x –3y 5x 2 –4y 2 +6y – 3 (–) (+) (–) (+) 2x 2 – 4xy +12y 2 + 5x – 9y + 3 140 MATHEMATICS Note that subtraction of a number is the same as addition of its additive inverse. Thus subtracting –3 is the same as adding +3. Similarly , subtracting 6y is the same as adding – 6y; subtracting – 4y 2 is the same as adding 4y 2 and so on. The signs in the third row written below each term in the second row help us in knowing which operation has to be performed. EXERCISE 9.1 1. Identify the terms, their coefficients for each of the following expressions. (i) 5xyz 2 – 3zy (ii) 1 + x + x 2 (iii) 4x 2 y 2 – 4x 2 y 2 z 2 + z 2 (iv) 3 – pq + qr – rp (v) 22 xy xy +- (vi) 0.3a – 0.6ab + 0.5b 2. Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories? x + y, 1000, x + x 2 + x 3 + x 4 , 7 + y + 5x, 2y – 3y 2 , 2y – 3y 2 + 4y 3 , 5x – 4y + 3xy, 4z – 15z 2 , ab + bc + cd + da, pqr, p 2 q + pq 2 , 2p + 2q 3. Add the following. (i) ab – bc, bc – ca, ca – ab (ii) a – b + ab, b – c + bc, c – a + ac (iii) 2p 2 q 2 – 3pq + 4, 5 + 7pq – 3p 2 q 2 (iv) l 2 + m 2 , m 2 + n 2 , n 2 + l 2 , 2lm + 2mn + 2nl 4. (a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3 (b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz (c) Subtract 4p 2 q – 3pq + 5pq 2 – 8p + 7q – 10 from 18 – 3p – 11q + 5pq – 2pq 2 + 5p 2 q 9.6 Multiplication of Algebraic Expressions: Introduction (i) Look at the following patterns of dots. Pattern of dots Total number of dots 4 × 9 5 × 7 ALGEBRAIC EXPRESSIONS AND IDENTITIES 141 m × n (m + 2) × (n + 3) (ii) Can you now think of similar other situations in which two algebraic expressions have to be multiplied? Ameena gets up. She says, “W e can think of area of a rectangle.” The area of a rectangle is l × b, where l is the length, and b is breadth. If the length of the rectangle is increased by 5 units, i.e., (l + 5) and breadth is decreased by 3 units , i.e., (b – 3) units, the area of the new rectangle will be (l + 5) × (b – 3). (iii) Can you think about volume? (The volume of a rectangular box is given by the product of its length, breadth and height). (iv) Sarita points out that when we buy things, we have to carry out multiplication. For example, if price of bananas per dozen = ` p and for the school picnic bananas needed = z dozens, then we have to pay = ` p × z Suppose, the price per dozen was less by ` 2 and the bananas needed were less by 4 dozens. Then, price of bananas per dozen = ` (p – 2) and bananas needed = (z – 4) dozens, Therefore, we would have to pay = ` (p – 2) × (z – 4) To find the area of a rectangle, we have to multiply algebraic expressions like l × b or (l + 5) × (b – 3). Here the number of rows is increased by 2, i.e., m + 2 and number of columns increased by 3, i.e., n + 3. To find the number of dots we have to multiply the expression for the number of rows by the expression for the number of columns.Read More

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