1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient of -5xy is -5 Coefficient of 6y 2 is 6. The value of the expression 10a 3 + 5a 2 15b -17ab when a = -1 and b = 2 is 10(-1) 3 + 5(-1) 2 15(2) 17(-1)(2) = -1 1) Find the value of the expression 6x 3 x 2 y 2 +8y 3 when x = -2 and y = 1. 2) What should be taken away from 15x 2 + 5xy 9y 2 to get -3x 2 + 6xy 9y 2 1) -44 2) 18x 2 xy 2. Algebraic Expressions Description An algebraic expression is a combination of variables and numerals separated by mathematical symbols. In the expression 6x 2-5x + 2y, there are three terms 6x 2, -5x and 2y where x and y are variables. The numerical factor accompanying a variable is known as coefficient. 6 is the coefficient of x 2, -5 is the coefficient of x and 2 is the coefficient of y. Polynomial: A polynomial is an algebraic expression containing one or more terms with variables Reflect and review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17 Examples of monomial: 7x 4, -10 Examples of binomial: 3y Teasers 1) Write the coefficient of all the terms in -17x 2 y + 4y 2 z 10z 2 x 2) a) Write a binomial using the variable x. b) Write a trinomial using the variable z. Answers 1) Coefficient of x 2 y = -17 Coefficient of y 2 z = 4 Coefficient of z 2 x = -10 2) Answers will vary. 1
having exponents as nonnegative integers. If an algebraic expression has only one term, it is called a monomial. An algebraic expression having two terms is a binomial. An algebraic expression having three terms is a trinomial. 4z, 2x 2 + 9 Examples of trinomial: x 2 y 2 + z 2, 2x 7y + 9 3. Like and Unlike Terms Description If the terms have same variable and same coefficient, then they are called like terms; otherwise they are called unlike terms. Reflect and review The like and unlike terms among 12a 2, - 15bc, 19c 2, 5a 2, - 10ac, 21bc, 17b 2, 21a 2 are Like terms: 12a 2, 5a 2, 21a 2 ; -15bc, 21bc Teasers 1) Pick the like and unlike terms from the following 3x 2, 5yz, - 19x 2, 18yz 2, - 25yz, 18y 2, - 10x 2, 16yz Answers 1) Like terms: 3x 2, -19x 2, - 10x 2 ; 5yz, - 25yz, 16yz Unlike terms: 18yz 2, 18y 2 4. Degree of a Polynomial Unlike terms: 19c 2, - 10ac, 17b 2 The degree of a polynomial is the highest power of the variable in the expression. The degree of the polynomial 8y 3 y 4 + 6y 2y 2 10 is 4. Since the highest power of the variable y is 4. 1) Find the degree of the polynomial 10a 5 21a 3-35 1) 5 5. Evaluation of an Algebraic Expression Description The value of the algebraic expression is obtained by replacing the given values of the variables in it and then calculating accordingly. Reflect and review The value of the expression 12ab 2 5bc 2 + 2ca 2 when a = 1, b = -1 and c = 2 is 12ab 2 5bc 2 + 2ca 2 = 12(1)(-1) 2 5(-1)(2) 2 + 2(2)(1) 2 = 12 + 20 + Teasers 1) Find the value of the expression 18x 2 y 2 + 10 y 2 z 2-20 x 2 z 2 when x = - 1, y = -2 and z = 1 2) Find the value of 15a 3 + 19b 3 + 9c 3 21 when a = 0, b = Answers 1) 92 2) -30 2
4 = 36 0 and c = -1 6. Addition and Subtraction of Algebraic Expressions The sum or difference is possible only for like terms but not for unlike terms. If all the like terms are of the same sign, then add them and the answer will have the same sign. If two like terms are of opposite sign, then subtract them and the answer will have the sign of the term having numerically bigger coefficient. Adding 3x 2 4xy and -5x 2 6xy + 7y 2 is as shown below 3x 2 4xy -5x 2 6xy + 7y 2-2x 2 10xy + 7y 2 Subtracting 7x 14xy + 19y from 16x + 21xy 11y is as shown below 7x 14xy + 19y 16x + 21xy 11y (-) (-) (+) -9x 35xy +30y 1) Add: 8pq 6qr 2 + 15pr, -17pq + 11qr 2 10pr, pq 5qr 2 3pr 2) Subtract: 16xy 9yz + 2xz from the sum of 2xy +20yz + 15xz and xy + 2yz 5xz 1) -8pq + 2pr 2) -15xy + 29yz +12xz 7. Multiplication of Algebraic Expressions two expressions, multiply each term of the first expression with each term of the second expression. These expressions can be monomials, binomials or trinomials. Multiplication of a Monomial by a Monomial 5a 2 and 16ab 2 : 5a 2 16ab 2 = 5 a a 16 a b b = (5 16) a a a b b = 80a 3 b 2 Find the product of these monomials. 1), 2) -4 abc, 5b 2, 9ac 5, 1) 2) -180a 2 b 3 c 6 ½ p, ¾ pqr and -48 r 2 : ½ p ¾ pqr -48 r 2 = ½ p ¾ p q r 3
(-48) r r = ½ ¾ (-48) p p q r r r = -18p 2 qr 3 Multiplication of Monomial with a Binomial or a Trinomial 5a and 10a 2 + 3b 2 : 5a (10a 2 + 3b 2 ) = 5a 10a 2 + 5a 3b 2 = 50a 3 + 15ab 2-12y 2 and ½ yz + ¼ z 3 : -12y 2+ (½ yz + ¼ z 3 ) = (-12y 2 ) ½ yz + (- 12y 2 ) ¼ z 3 = -6y 3 z 3y 2 z 3 Evaluation of Algebraic Expression after Simplification To find the value of 5p(2p 6q) + 2q(-10p + 3q) when p = 2 and q = 1: The product of 5p and (2p 6q) is 5p(2p 6q) = 5p 2p + 5p (-6q) = 10p 2 30pq Similarly, the product of 2q and (-10p + 3q) is 2q(-10p + 3q) = 2q (-10p) + 2q 3q = -20pq + 6q 2 Hence, 5p(2p 6q) + 2q(-10p + 3q) = (10p 2 30pq) + (-20pq + 6q 2 ) = 10p 2 50pq +6q 2 Putting p = 2 and q = 1 in the expression, we get 1) Find the product of 12a, -15ab + 2 bc 10 ca 2) Simplify a 2 (b c) + b 2 (c a) + c 2 (a b) 1) Evaluate 8y(y 2 2y + 1) 6y 2 (2y 9) for y = -1 2) Subtract 5x(2x 3y) from the sum of 4x(y 7x) and 2x(-x - y) 1) -180a 2 b + 24abc - 120a 2 c 2) a 2 b - a 2 c + b 2 c - b 2 a + c 2 a - c 2 b 1) 34 2) -40x 2 13xy 4
10(2) 2 50(2)(1) + 6(1) 2 = 40 100 + 6 = 54 Multiplication of a Binomial and a Binomial (5x 3y) and (6y 7x): (5x 3y)(6y 7x) = 5x(6y 7x) - 3y(6y 7x) = 30xy 35x 2 18y 2 + 21xy = -35x 2 + 51xy 18y 2 Evaluate 1) (p + q)( p - q) + (q + r)(q - r) + (r + p)(r - p) 2) (a + b)(a 2 ab + b 2 ) + (a - b)(a 2 + ab + b 2 ) 1) 0 2) 2a 3 (10a 6b) and (-a + 2b + 7c): (10a 6b)(-a + 2b +7c) = 10a(-a + 2b +7c) -6b(- a + 2b + 7c) = -10a 2 + 20ab +70ac + 6ab 12b 2 42bc = -10a 2 + 26ab -12 b 2 + 70ac 42bc 8. Identities An equality, which is true for all values of the variable, is called an identity. Standard Identities (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2-2ab + b 2 (a + b)(a b) = a 2 b 2 (x + a)(x + b) = x 2 + (a + b)x + ab (2a + 5b) 2 can be identity, (a + b) 2 = a 2 + 2ab + b 2 (2a + 5b) 2 = (2a) 2 + 2(2a)(5b) + (5b) 2 = 4a 2 + 20ab + 25b 2 (96) 2 can be identity, (a - b) 2 = a 2-2ab + b 2 (96) 2 = (100 4) 2 = Evaluate the following using the appropriate identities. 1) (1012) 2 2) (-5a + 3b) 2 3) (-10p + 3q)(10p + 3q) 4) (a 0.5)(a 1.8) 5) 56 2-44 2 1) 1024144 2) 25a 2 30ab + 9b 2 3) 9q 2 100p 2 4) a 2-2.3a + 0.9 5) 1200 5
(100) 2 2(100)(4) + (4) 2 = 10000 800 + 16 = 9216 62 58 can be identity, (a + b)(a b) = a 2 b 2 62 58 = (60 + 2)(60 2) = (60) 2 (2) 2 = 3600 4 = 3596 (a + 7)(a 10) can be identity (x +a )(x + b) = x 2 + (a + b)x + ab (a + 7)(a 10) = a 2 + [(7) + (-10)]a + (7)(-10) = a 2 3a 70 6