Page 1 Points to Remember : Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known as literal numbers or simply literals. Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and division. Operations on Literals and Numbers : 1. Addition : (i) The sum of literal x and a number 5 is x + 5 (ii) y more than x is written as x + y (iii) For any literals a, b, c we have a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c) 2. Subtraction : (i) 5 less then a literal x is x – 5 ; (ii) y less than x is x – y. 3. Multiplication : (i) 4 times x is 4 × x, written as 4x. (ii) The product of x and y is x × y, written as xy. (iii) For any literals a, b, c we have : a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c) = ab + ac. 4. Division : (i) x divided by y is written as x y (ii) x divided by 5 is x 5 . 10 divided by x is 10 x Powers of a literal x x is written as x 2 , called x squared. x x x is written as x 3 , called x cubed x x x x is written as x 4 , called x raised to the power 4 and so on In x 4 , x is called the base and 4 the exponent or index. Page 2 Points to Remember : Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known as literal numbers or simply literals. Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and division. Operations on Literals and Numbers : 1. Addition : (i) The sum of literal x and a number 5 is x + 5 (ii) y more than x is written as x + y (iii) For any literals a, b, c we have a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c) 2. Subtraction : (i) 5 less then a literal x is x – 5 ; (ii) y less than x is x – y. 3. Multiplication : (i) 4 times x is 4 × x, written as 4x. (ii) The product of x and y is x × y, written as xy. (iii) For any literals a, b, c we have : a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c) = ab + ac. 4. Division : (i) x divided by y is written as x y (ii) x divided by 5 is x 5 . 10 divided by x is 10 x Powers of a literal x x is written as x 2 , called x squared. x x x is written as x 3 , called x cubed x x x x is written as x 4 , called x raised to the power 4 and so on In x 4 , x is called the base and 4 the exponent or index. ( ) EXERCISE 8 A Q. 1. Write the following using literals, numbers and signs of basic operations: (i) x increased by 12 (ii) y decreased by 7 (iii) The difference of a and b, when a > b (iv) The product of x and y added to their sum (v) One third of x multiplied by the sum of a and b (vi) 5 times x added to 7 times y (vii) Sum of x and quotient of y by 5 (viii) x taken away from 4 (ix) 2 less than the quotient of x by y (x) x multiplied by itself (xi) Twice x increased by y (xii) Thrice x added to y squared (xiii) x minus twice y (xiv) x cubed less than y cubed. (xv) Quotient of x by 8 is multiplied by y. Sol. (i) x + 12 (ii) y – 7 (iii) a – b (iv) (x + y) + xy (v) 1 3 x a b ( ) (vi) 7y + 5x (vii) x y 5 (viii) 4 – x (ix) x y 2 (x) x 2 (xi) 2x + y (xii) y 2 + 3x (xiii) x – 2y (xiv) y 3 – x 3 (xv) x y 8 Q. 2. Ranjit scores 80 marks in English and x marks in Hindi. What is his total score in the two subjects ? Sol. Marks scored in English = 80 Marks scored in Hindi = x Total score in the two subjects = 80 + x Q. 3. Write the following in exponential form : (i) b b b ......... 15 times (ii) y y y ............. 20 times (iii) 14 a a a a b b b (iv) 6 x x y y (v) 3 z z z y y x Sol. We can write : (i) b b b ................. 15 times = b 15 (ii) y y y .................. 20 times = y 20 (iii) 14 a a a a b b b = 14a 4 b 3 (iv) 6 x x y y = 6x 2 y 2 (v) 3 z z z y y x = 3z 3 y 2 x Q. 4. Write down the following in product form : (i) x 2 y 4 (ii) 6y 5 (iii) 9xy 2 z (iv) 10a 3 b 3 c 3 Sol. We can write : (i) x 2 y 4 = x x y y y y (ii) 6y 5 = 6 y y y y y (iii) 9xy 2 z = 9 x y y z (iv) 10a 3 b 3 c 3 = 10 a a a b b b c c c Algebraic Expressions Constant. A symbol having a fixed numerical value is called a constant. Variable. A symbol which takes on various numerical values is called a variable. Algebraic Expression. A combination of constants and variables, connected by the symbols +, –, and is called an algebraic expression. The several parts of the expression separated by the sign + or – are called the ‘terms’ of the expression. Various types of algebraic expressions are : (i) Monomials. An expression containing only one term is called a monomial. Page 3 Points to Remember : Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known as literal numbers or simply literals. Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and division. Operations on Literals and Numbers : 1. Addition : (i) The sum of literal x and a number 5 is x + 5 (ii) y more than x is written as x + y (iii) For any literals a, b, c we have a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c) 2. Subtraction : (i) 5 less then a literal x is x – 5 ; (ii) y less than x is x – y. 3. Multiplication : (i) 4 times x is 4 × x, written as 4x. (ii) The product of x and y is x × y, written as xy. (iii) For any literals a, b, c we have : a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c) = ab + ac. 4. Division : (i) x divided by y is written as x y (ii) x divided by 5 is x 5 . 10 divided by x is 10 x Powers of a literal x x is written as x 2 , called x squared. x x x is written as x 3 , called x cubed x x x x is written as x 4 , called x raised to the power 4 and so on In x 4 , x is called the base and 4 the exponent or index. ( ) EXERCISE 8 A Q. 1. Write the following using literals, numbers and signs of basic operations: (i) x increased by 12 (ii) y decreased by 7 (iii) The difference of a and b, when a > b (iv) The product of x and y added to their sum (v) One third of x multiplied by the sum of a and b (vi) 5 times x added to 7 times y (vii) Sum of x and quotient of y by 5 (viii) x taken away from 4 (ix) 2 less than the quotient of x by y (x) x multiplied by itself (xi) Twice x increased by y (xii) Thrice x added to y squared (xiii) x minus twice y (xiv) x cubed less than y cubed. (xv) Quotient of x by 8 is multiplied by y. Sol. (i) x + 12 (ii) y – 7 (iii) a – b (iv) (x + y) + xy (v) 1 3 x a b ( ) (vi) 7y + 5x (vii) x y 5 (viii) 4 – x (ix) x y 2 (x) x 2 (xi) 2x + y (xii) y 2 + 3x (xiii) x – 2y (xiv) y 3 – x 3 (xv) x y 8 Q. 2. Ranjit scores 80 marks in English and x marks in Hindi. What is his total score in the two subjects ? Sol. Marks scored in English = 80 Marks scored in Hindi = x Total score in the two subjects = 80 + x Q. 3. Write the following in exponential form : (i) b b b ......... 15 times (ii) y y y ............. 20 times (iii) 14 a a a a b b b (iv) 6 x x y y (v) 3 z z z y y x Sol. We can write : (i) b b b ................. 15 times = b 15 (ii) y y y .................. 20 times = y 20 (iii) 14 a a a a b b b = 14a 4 b 3 (iv) 6 x x y y = 6x 2 y 2 (v) 3 z z z y y x = 3z 3 y 2 x Q. 4. Write down the following in product form : (i) x 2 y 4 (ii) 6y 5 (iii) 9xy 2 z (iv) 10a 3 b 3 c 3 Sol. We can write : (i) x 2 y 4 = x x y y y y (ii) 6y 5 = 6 y y y y y (iii) 9xy 2 z = 9 x y y z (iv) 10a 3 b 3 c 3 = 10 a a a b b b c c c Algebraic Expressions Constant. A symbol having a fixed numerical value is called a constant. Variable. A symbol which takes on various numerical values is called a variable. Algebraic Expression. A combination of constants and variables, connected by the symbols +, –, and is called an algebraic expression. The several parts of the expression separated by the sign + or – are called the ‘terms’ of the expression. Various types of algebraic expressions are : (i) Monomials. An expression containing only one term is called a monomial. Various types of algebraic expressions are : (i) Monomials. An expression containing only one term is called a monomial. (ii) Binomials : An expression containing two terms is called a Binomial. (iii) Trinomials : An expression containing three terms is called a trinomial. (iv) Quadrinomials : An expression containing four terms is called a quadrinomial. (v) Polynomials : An expression containing two or more terms is known as a polynomial. Factors : When two or more numbers and literals are multiplied then each one of them is called a factor of the product. Coefficients : In a product of numbers and literals, any of the factors is called the coefficient of the product of other factors. Constant Term : A term of the expression having no literal factor is called a constant term. Like Terms : The terms having same literal factors are called like or similar terms. Unlike Terms : The terms not having same literal factors are called unlike or dissimilar terms. ( ) EXERCISE 8 B Q. 1. If a = 2, b = 3, find the value of : (i) a + b (ii) a 2 + ab (iii) ab – a 2 (iv) 2a – 3b (v) 5a 2 – 2ab (vi) a 3 – b 3 Sol. (i) Substituting a = 2 and b = 3 in the given expression, we get : a + b = 2 + 3 = 5 (ii) Substituting a = 2 and b = 3 in the given expression, we get : a 2 + ab = (2) 2 + 2 × 3 = 4 + 6 = 10 (iii) Substituting a = 2 and b = 3 in the given expression, we get : ab – a 2 = 2 × 3 – (2) 2 = 6 – 4 = 2 (iv) Substituting a = 2 and b = 3 in the given expression, we get : 2a – 3b = 2 × 2 – 3 3 = 4 – 9 = – 5 (v) Substituting a = 2 and b = 3 in the given expression, we get : 5a 2 – 2ab = 5 × (2) 2 – 2 × 2 × 3 = 5 × 4 – 4 3 = 20 – 12 = 8 (vi) Substituting a = 2 and b = 3 in the given expression, we get : a 3 – b 3 = (2) 3 – (3) 3 = 2 × 2 × 2 – 3 × 3 × 3 = 8 – 27 = – 19 Q. 2. If x = 1, y = 2 and z = 5, find the value of : (i) 3x – 2y + 4z (ii) x 2 + y 2 + z 2 (iii) 2x 2 – 3y 2 + z 2 (iv) xy + yz – zx (v) 2x 2 y – 5yz + xy 2 (vi) x 3 – y 3 – z 3 Sol. (i) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 3x – 2y + 4z = 3 × 1 – 2 × 2 + 4 × 5 = 3 – 4 + 20 = 23 – 4 = 19 (ii) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : x 2 + y 2 + z 2 = (1) 2 + (2) 2 + (5) 2 = 1 + 4 + 25 = 30 (iii) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 2x 2 – 3y 2 + z 2 = 2 (1) 2 – 3 (2) 2 + (5) 2 = 2 1 – 3 4 + 25 = 2 – 12 + 25 = 27 – 12 = 15. (iv) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : Page 4 Points to Remember : Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known as literal numbers or simply literals. Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and division. Operations on Literals and Numbers : 1. Addition : (i) The sum of literal x and a number 5 is x + 5 (ii) y more than x is written as x + y (iii) For any literals a, b, c we have a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c) 2. Subtraction : (i) 5 less then a literal x is x – 5 ; (ii) y less than x is x – y. 3. Multiplication : (i) 4 times x is 4 × x, written as 4x. (ii) The product of x and y is x × y, written as xy. (iii) For any literals a, b, c we have : a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c) = ab + ac. 4. Division : (i) x divided by y is written as x y (ii) x divided by 5 is x 5 . 10 divided by x is 10 x Powers of a literal x x is written as x 2 , called x squared. x x x is written as x 3 , called x cubed x x x x is written as x 4 , called x raised to the power 4 and so on In x 4 , x is called the base and 4 the exponent or index. ( ) EXERCISE 8 A Q. 1. Write the following using literals, numbers and signs of basic operations: (i) x increased by 12 (ii) y decreased by 7 (iii) The difference of a and b, when a > b (iv) The product of x and y added to their sum (v) One third of x multiplied by the sum of a and b (vi) 5 times x added to 7 times y (vii) Sum of x and quotient of y by 5 (viii) x taken away from 4 (ix) 2 less than the quotient of x by y (x) x multiplied by itself (xi) Twice x increased by y (xii) Thrice x added to y squared (xiii) x minus twice y (xiv) x cubed less than y cubed. (xv) Quotient of x by 8 is multiplied by y. Sol. (i) x + 12 (ii) y – 7 (iii) a – b (iv) (x + y) + xy (v) 1 3 x a b ( ) (vi) 7y + 5x (vii) x y 5 (viii) 4 – x (ix) x y 2 (x) x 2 (xi) 2x + y (xii) y 2 + 3x (xiii) x – 2y (xiv) y 3 – x 3 (xv) x y 8 Q. 2. Ranjit scores 80 marks in English and x marks in Hindi. What is his total score in the two subjects ? Sol. Marks scored in English = 80 Marks scored in Hindi = x Total score in the two subjects = 80 + x Q. 3. Write the following in exponential form : (i) b b b ......... 15 times (ii) y y y ............. 20 times (iii) 14 a a a a b b b (iv) 6 x x y y (v) 3 z z z y y x Sol. We can write : (i) b b b ................. 15 times = b 15 (ii) y y y .................. 20 times = y 20 (iii) 14 a a a a b b b = 14a 4 b 3 (iv) 6 x x y y = 6x 2 y 2 (v) 3 z z z y y x = 3z 3 y 2 x Q. 4. Write down the following in product form : (i) x 2 y 4 (ii) 6y 5 (iii) 9xy 2 z (iv) 10a 3 b 3 c 3 Sol. We can write : (i) x 2 y 4 = x x y y y y (ii) 6y 5 = 6 y y y y y (iii) 9xy 2 z = 9 x y y z (iv) 10a 3 b 3 c 3 = 10 a a a b b b c c c Algebraic Expressions Constant. A symbol having a fixed numerical value is called a constant. Variable. A symbol which takes on various numerical values is called a variable. Algebraic Expression. A combination of constants and variables, connected by the symbols +, –, and is called an algebraic expression. The several parts of the expression separated by the sign + or – are called the ‘terms’ of the expression. Various types of algebraic expressions are : (i) Monomials. An expression containing only one term is called a monomial. Various types of algebraic expressions are : (i) Monomials. An expression containing only one term is called a monomial. (ii) Binomials : An expression containing two terms is called a Binomial. (iii) Trinomials : An expression containing three terms is called a trinomial. (iv) Quadrinomials : An expression containing four terms is called a quadrinomial. (v) Polynomials : An expression containing two or more terms is known as a polynomial. Factors : When two or more numbers and literals are multiplied then each one of them is called a factor of the product. Coefficients : In a product of numbers and literals, any of the factors is called the coefficient of the product of other factors. Constant Term : A term of the expression having no literal factor is called a constant term. Like Terms : The terms having same literal factors are called like or similar terms. Unlike Terms : The terms not having same literal factors are called unlike or dissimilar terms. ( ) EXERCISE 8 B Q. 1. If a = 2, b = 3, find the value of : (i) a + b (ii) a 2 + ab (iii) ab – a 2 (iv) 2a – 3b (v) 5a 2 – 2ab (vi) a 3 – b 3 Sol. (i) Substituting a = 2 and b = 3 in the given expression, we get : a + b = 2 + 3 = 5 (ii) Substituting a = 2 and b = 3 in the given expression, we get : a 2 + ab = (2) 2 + 2 × 3 = 4 + 6 = 10 (iii) Substituting a = 2 and b = 3 in the given expression, we get : ab – a 2 = 2 × 3 – (2) 2 = 6 – 4 = 2 (iv) Substituting a = 2 and b = 3 in the given expression, we get : 2a – 3b = 2 × 2 – 3 3 = 4 – 9 = – 5 (v) Substituting a = 2 and b = 3 in the given expression, we get : 5a 2 – 2ab = 5 × (2) 2 – 2 × 2 × 3 = 5 × 4 – 4 3 = 20 – 12 = 8 (vi) Substituting a = 2 and b = 3 in the given expression, we get : a 3 – b 3 = (2) 3 – (3) 3 = 2 × 2 × 2 – 3 × 3 × 3 = 8 – 27 = – 19 Q. 2. If x = 1, y = 2 and z = 5, find the value of : (i) 3x – 2y + 4z (ii) x 2 + y 2 + z 2 (iii) 2x 2 – 3y 2 + z 2 (iv) xy + yz – zx (v) 2x 2 y – 5yz + xy 2 (vi) x 3 – y 3 – z 3 Sol. (i) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 3x – 2y + 4z = 3 × 1 – 2 × 2 + 4 × 5 = 3 – 4 + 20 = 23 – 4 = 19 (ii) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : x 2 + y 2 + z 2 = (1) 2 + (2) 2 + (5) 2 = 1 + 4 + 25 = 30 (iii) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 2x 2 – 3y 2 + z 2 = 2 (1) 2 – 3 (2) 2 + (5) 2 = 2 1 – 3 4 + 25 = 2 – 12 + 25 = 27 – 12 = 15. (iv) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : xy + yz – zx = 1 2 + 2 5 – 5 1 = 2 + 10 – 5 = 12 – 5 = 7. (v) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 2x 2 y – 5yz + xy 2 = 2 (1) 2 2 – 5 2 5 + 1 (2) 2 = 2 1 2 –10 5 1 4 = 4 – 50 + 4 = 8 – 50 = – 42 (vi) Substitutng x = 1, y = 2 and z = 5 in the given expression, we get : x 3 – y 3 – z 3 = (1) 3 – (2) 3 – (5) 3 = (1 1 1) – (2 2 2) – (5 5 5) = 1 – 8 – 125 = 1 – 133 = – 132 Q. 3. If p = – 2, q = – 1 and r = 3, find the value of : (i) p 2 + q 2 – r 2 (ii) 2p 2 – q 2 + 3r 2 (iii) p – q – r (iv) p 3 + q 3 + r 3 + 3pqr (v) 3p 2 q + 5pq 2 + 2pqr (vi) p 4 + q 4 – r 4 Sol. (i) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p 2 + q 2 – r 2 = (– 2) 2 + (–1) 2 – (3) 2 = 4 + 1 – 9 = 5 – 9 = – 4 (ii) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : 2p 2 – q 2 + 3r 2 = 2 × (– 2) 2 – (– 1) 2 + 3 × (3) 2 = 2 × 4 – 1 + 3 × 9 = 8 – 1 + 27 = 34 (iii) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p – q – r = (– 2) – (– 1) – 3 = – 2 + 1 – 3 = – 4 (iv) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p 3 + q 3 + r 3 + 3pqr = (– 2) 3 + (– 1) 3 + (3) 3 + 3 (– 2) × (– 1) 3 = (– 8) + (– 1) + 27 + 18 = – 8 –1 + 27 + 18 = – 9 + 45 = 36 (v) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : 3p 2 q + 5pq 2 + 2pqr = 3 (–2) 2 (– 1) + 5 × (– 2) × (– 1) 2 + 2 (– 2) (– 1) 3 = 3 4 (– 1) + 5 (– 2) 1 + 12 = – 12 – 10 + 12 = – 10 (vi) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p 4 + q 4 – r 4 = (– 2) 4 + (– 1) 4 – (3) 4 = 16 + 1 – 81 = 17 – 81 = – 64 Q. 4. Write the coefficient of : (i) x in 13x (ii) y in – 5y (iii) a in 6ab (iv) z in – 7xz (v) p in – 2pqr (vi) y 2 in 8xy 2 z (vii) x 3 in x 3 (viii) x 2 in – x 2 Sol. (i) The coefficient of x in 13x is 13 (ii) The coefficient of y in – 5y is – 5 (iii) The coefficient of a in 6ab is 6b (iv) The coefficient of z in – 7xz is – 7x (v) The coefficient of p in – 2pqr is – 2qr (vi) The coefficient of y 2 in 8xy 2 z is 8xz (vii) The coefficient of x 3 in x 3 is 1 (viii) The coefficient of x 2 in – x 2 is –1 Q. 5. Write the numerical coefficient of : (i) ab (ii) – 6bc (iii) 7xyz (iv) – 2x 3 y 3 z Sol. (i) The numerical coefficeint of ab is 1 (ii) The numerical coefficient of – 6bc is – 6 (iii) The numerical coefficient of 7xyz is 7 Page 5 Points to Remember : Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known as literal numbers or simply literals. Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and division. Operations on Literals and Numbers : 1. Addition : (i) The sum of literal x and a number 5 is x + 5 (ii) y more than x is written as x + y (iii) For any literals a, b, c we have a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c) 2. Subtraction : (i) 5 less then a literal x is x – 5 ; (ii) y less than x is x – y. 3. Multiplication : (i) 4 times x is 4 × x, written as 4x. (ii) The product of x and y is x × y, written as xy. (iii) For any literals a, b, c we have : a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c) = ab + ac. 4. Division : (i) x divided by y is written as x y (ii) x divided by 5 is x 5 . 10 divided by x is 10 x Powers of a literal x x is written as x 2 , called x squared. x x x is written as x 3 , called x cubed x x x x is written as x 4 , called x raised to the power 4 and so on In x 4 , x is called the base and 4 the exponent or index. ( ) EXERCISE 8 A Q. 1. Write the following using literals, numbers and signs of basic operations: (i) x increased by 12 (ii) y decreased by 7 (iii) The difference of a and b, when a > b (iv) The product of x and y added to their sum (v) One third of x multiplied by the sum of a and b (vi) 5 times x added to 7 times y (vii) Sum of x and quotient of y by 5 (viii) x taken away from 4 (ix) 2 less than the quotient of x by y (x) x multiplied by itself (xi) Twice x increased by y (xii) Thrice x added to y squared (xiii) x minus twice y (xiv) x cubed less than y cubed. (xv) Quotient of x by 8 is multiplied by y. Sol. (i) x + 12 (ii) y – 7 (iii) a – b (iv) (x + y) + xy (v) 1 3 x a b ( ) (vi) 7y + 5x (vii) x y 5 (viii) 4 – x (ix) x y 2 (x) x 2 (xi) 2x + y (xii) y 2 + 3x (xiii) x – 2y (xiv) y 3 – x 3 (xv) x y 8 Q. 2. Ranjit scores 80 marks in English and x marks in Hindi. What is his total score in the two subjects ? Sol. Marks scored in English = 80 Marks scored in Hindi = x Total score in the two subjects = 80 + x Q. 3. Write the following in exponential form : (i) b b b ......... 15 times (ii) y y y ............. 20 times (iii) 14 a a a a b b b (iv) 6 x x y y (v) 3 z z z y y x Sol. We can write : (i) b b b ................. 15 times = b 15 (ii) y y y .................. 20 times = y 20 (iii) 14 a a a a b b b = 14a 4 b 3 (iv) 6 x x y y = 6x 2 y 2 (v) 3 z z z y y x = 3z 3 y 2 x Q. 4. Write down the following in product form : (i) x 2 y 4 (ii) 6y 5 (iii) 9xy 2 z (iv) 10a 3 b 3 c 3 Sol. We can write : (i) x 2 y 4 = x x y y y y (ii) 6y 5 = 6 y y y y y (iii) 9xy 2 z = 9 x y y z (iv) 10a 3 b 3 c 3 = 10 a a a b b b c c c Algebraic Expressions Constant. A symbol having a fixed numerical value is called a constant. Variable. A symbol which takes on various numerical values is called a variable. Algebraic Expression. A combination of constants and variables, connected by the symbols +, –, and is called an algebraic expression. The several parts of the expression separated by the sign + or – are called the ‘terms’ of the expression. Various types of algebraic expressions are : (i) Monomials. An expression containing only one term is called a monomial. Various types of algebraic expressions are : (i) Monomials. An expression containing only one term is called a monomial. (ii) Binomials : An expression containing two terms is called a Binomial. (iii) Trinomials : An expression containing three terms is called a trinomial. (iv) Quadrinomials : An expression containing four terms is called a quadrinomial. (v) Polynomials : An expression containing two or more terms is known as a polynomial. Factors : When two or more numbers and literals are multiplied then each one of them is called a factor of the product. Coefficients : In a product of numbers and literals, any of the factors is called the coefficient of the product of other factors. Constant Term : A term of the expression having no literal factor is called a constant term. Like Terms : The terms having same literal factors are called like or similar terms. Unlike Terms : The terms not having same literal factors are called unlike or dissimilar terms. ( ) EXERCISE 8 B Q. 1. If a = 2, b = 3, find the value of : (i) a + b (ii) a 2 + ab (iii) ab – a 2 (iv) 2a – 3b (v) 5a 2 – 2ab (vi) a 3 – b 3 Sol. (i) Substituting a = 2 and b = 3 in the given expression, we get : a + b = 2 + 3 = 5 (ii) Substituting a = 2 and b = 3 in the given expression, we get : a 2 + ab = (2) 2 + 2 × 3 = 4 + 6 = 10 (iii) Substituting a = 2 and b = 3 in the given expression, we get : ab – a 2 = 2 × 3 – (2) 2 = 6 – 4 = 2 (iv) Substituting a = 2 and b = 3 in the given expression, we get : 2a – 3b = 2 × 2 – 3 3 = 4 – 9 = – 5 (v) Substituting a = 2 and b = 3 in the given expression, we get : 5a 2 – 2ab = 5 × (2) 2 – 2 × 2 × 3 = 5 × 4 – 4 3 = 20 – 12 = 8 (vi) Substituting a = 2 and b = 3 in the given expression, we get : a 3 – b 3 = (2) 3 – (3) 3 = 2 × 2 × 2 – 3 × 3 × 3 = 8 – 27 = – 19 Q. 2. If x = 1, y = 2 and z = 5, find the value of : (i) 3x – 2y + 4z (ii) x 2 + y 2 + z 2 (iii) 2x 2 – 3y 2 + z 2 (iv) xy + yz – zx (v) 2x 2 y – 5yz + xy 2 (vi) x 3 – y 3 – z 3 Sol. (i) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 3x – 2y + 4z = 3 × 1 – 2 × 2 + 4 × 5 = 3 – 4 + 20 = 23 – 4 = 19 (ii) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : x 2 + y 2 + z 2 = (1) 2 + (2) 2 + (5) 2 = 1 + 4 + 25 = 30 (iii) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 2x 2 – 3y 2 + z 2 = 2 (1) 2 – 3 (2) 2 + (5) 2 = 2 1 – 3 4 + 25 = 2 – 12 + 25 = 27 – 12 = 15. (iv) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : xy + yz – zx = 1 2 + 2 5 – 5 1 = 2 + 10 – 5 = 12 – 5 = 7. (v) Substituting x = 1, y = 2 and z = 5 in the given expression, we get : 2x 2 y – 5yz + xy 2 = 2 (1) 2 2 – 5 2 5 + 1 (2) 2 = 2 1 2 –10 5 1 4 = 4 – 50 + 4 = 8 – 50 = – 42 (vi) Substitutng x = 1, y = 2 and z = 5 in the given expression, we get : x 3 – y 3 – z 3 = (1) 3 – (2) 3 – (5) 3 = (1 1 1) – (2 2 2) – (5 5 5) = 1 – 8 – 125 = 1 – 133 = – 132 Q. 3. If p = – 2, q = – 1 and r = 3, find the value of : (i) p 2 + q 2 – r 2 (ii) 2p 2 – q 2 + 3r 2 (iii) p – q – r (iv) p 3 + q 3 + r 3 + 3pqr (v) 3p 2 q + 5pq 2 + 2pqr (vi) p 4 + q 4 – r 4 Sol. (i) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p 2 + q 2 – r 2 = (– 2) 2 + (–1) 2 – (3) 2 = 4 + 1 – 9 = 5 – 9 = – 4 (ii) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : 2p 2 – q 2 + 3r 2 = 2 × (– 2) 2 – (– 1) 2 + 3 × (3) 2 = 2 × 4 – 1 + 3 × 9 = 8 – 1 + 27 = 34 (iii) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p – q – r = (– 2) – (– 1) – 3 = – 2 + 1 – 3 = – 4 (iv) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p 3 + q 3 + r 3 + 3pqr = (– 2) 3 + (– 1) 3 + (3) 3 + 3 (– 2) × (– 1) 3 = (– 8) + (– 1) + 27 + 18 = – 8 –1 + 27 + 18 = – 9 + 45 = 36 (v) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : 3p 2 q + 5pq 2 + 2pqr = 3 (–2) 2 (– 1) + 5 × (– 2) × (– 1) 2 + 2 (– 2) (– 1) 3 = 3 4 (– 1) + 5 (– 2) 1 + 12 = – 12 – 10 + 12 = – 10 (vi) Substituting p = – 2, q = – 1 and r = 3 in the given expression, we get : p 4 + q 4 – r 4 = (– 2) 4 + (– 1) 4 – (3) 4 = 16 + 1 – 81 = 17 – 81 = – 64 Q. 4. Write the coefficient of : (i) x in 13x (ii) y in – 5y (iii) a in 6ab (iv) z in – 7xz (v) p in – 2pqr (vi) y 2 in 8xy 2 z (vii) x 3 in x 3 (viii) x 2 in – x 2 Sol. (i) The coefficient of x in 13x is 13 (ii) The coefficient of y in – 5y is – 5 (iii) The coefficient of a in 6ab is 6b (iv) The coefficient of z in – 7xz is – 7x (v) The coefficient of p in – 2pqr is – 2qr (vi) The coefficient of y 2 in 8xy 2 z is 8xz (vii) The coefficient of x 3 in x 3 is 1 (viii) The coefficient of x 2 in – x 2 is –1 Q. 5. Write the numerical coefficient of : (i) ab (ii) – 6bc (iii) 7xyz (iv) – 2x 3 y 3 z Sol. (i) The numerical coefficeint of ab is 1 (ii) The numerical coefficient of – 6bc is – 6 (iii) The numerical coefficient of 7xyz is 7 (iv) The numerical coefficient of – 2x 3 y 3 z is – 2. Q. 6. Write the constant term of : (i) 3x 2 + 5x + 8 (ii) 2x 2 – 9 (iii) 4 5 3 5 2 y y (iv) z z z 3 2 2 8 3 Sol. (i) The constant term is 8 (ii) The constant term is – 9 (iii) The constant term is 3 5 (iv) The constant term is 8 3 Q. 7. Identify the monomials, binomials and trinomials in the following : (i) – 2xyz (ii) 5 + 7x 3 y 3 z 3 (iii) – 5x 3 (iv) a + b – 2c (v) xy + yz – zx (vi) x 5 (vii) ax 3 + bx 2 + cx + d (viii) – 14 (ix) 2x + 1 Sol. (i) The given expression contains only one term, so it is monimial. (ii) The given expression contains only two terms, so it is binomial. (iii) The given expression contains only one term, so it is monomial. (iv) The given expression contains three terms, so it is trinomial. (v) The given expression contains three terms, so it is trinomial. (vi) The given expression contains only one term, so it is monomial. (vii) The given expression contains four terms, so it is none of monomial, binomial and trinomial. (viii) The given expression contains only one term so it is monomial. (ix) The given expression contains two terms, so it is binomial. Q. 8. Write all the terms of the algebraic expressions : (i) 4x 5 – 6y 4 + 7x 2 y – 9 (ii) 9x 3 – 5z 4 + 7x 3 y – xyz Sol. (i) The terms of the given expression 4x 5 – 6y 4 + 7x 2 y – 9 are : 4x 5 , – 6y 4 , 7x 3 y, – 9 (ii) The terms of the given expression 9x 3 – 5z 4 + 7x 3 y – xyz are : 9x 3 , – 5z 4 , 7x 3 y, – xyz. Q. 9. Identify like terms in the following : (i) a 2 , b 2 , – 2a 2 , c 2 , 4a (ii) 3x, 4xy, – yz, 1 2 zy (iii) – 2xy 2 , x 2 y, 5y 2 x, x 2 z (iv) abc, ab 2 c, acb 2 , c 2 ab, b 2 ac, a 2 bc, cab 2 Sol. (i) We have : a 2 , b 2 , – 2a 2 , c 2 , 4a Here like terms are a 2 , – 2a 2 (ii) We have : 3x, 4xy, –yz, 1 2 zy Here like terms are yz zy , 1 2 (iii) We have : – 2xy 2 , x 2 y, 5y 2 x, x 2 z Here like terms are –2xy 2 , 5y 2 x (iv) We have : abc, ab 2 c, acb 2 , c 2 ab, b 2 ac, a 2 bc, cab 2 Here like terms are ab 2 c, acb 2 , b 2 ac, cab 2 . Operations on Algebraic Expressions 1. Addition of Algebraic Expressions The sum of several like terms is another like term whose coefficient is the sum of the coefficients of the like terms. Column Method. In this method, each expression is written in a separate row such that their like terms are arranged one below the other in a column. Then addition or subtraction of the terms is done columnwise.Read More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

191 videos|220 docs|43 tests