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ALGEBRA E ALGEBRA ALGEBRAIC IDENTITIES 6+2x - مد (2) = X -- = = 6. The HCF and LCM of two expressions are respec- tively (x + 3) and (x2 - 7x + 6). If one of the polyno- mials is x + 2x-3, the other polynomial is (1) x2 - X+6 (3) x2 - 2x + 8 (4) x2 + x-6 Sol. (4) p(x) x g(x) = HCF LCM = (x + 2x - 3) x g(x) = (x + 3) x (x3 - 7x + 6) (x + 3)(x3 - 7x + 6) :: Second expression = x2 + 2x - 3 (x+3)(x - 1)(x - 2)(x + 3) (x - 1)(x+3) (x + 3) (x-2) = x2 + x-6 7. If HCF and LCM of two terms x and y are a and b respectively and x + y = a + b, then x + y2 = ? (1) a? - 62 (2) 2a + b2 (3) a2 + b (4) a² + 2b2 Sol. (3) :: p(x) x (x) = LCM HCF cao xxy= bx a .. xy = ab and, x + y = a + b (Given) Squaring both sides, . (x + y)2 = (a + b)2 or, x2 + y2 + 2xy = a + b2 + 2ab or, x2 + y2 + 2ab = a2 + b2 + 2ab(:: xy = ab) :. x2 + y2 = a + b2 An algebraic identity is an algebraic equation which is true for all values of the variable (s). Important Formulae 1. (a + b)2 = a® + 2ab + b2 2. (a - b)2 = a - 2ab + b2 3. (a + b)2 = (a - b)2 + 4ab 4. (a - b)2 = (a + b)2 – 4ab 5. (a + b)3 = a + b3 + 3ab (a + b) 6. (a - b)3 = a - b - 3ab (a - b) 7. (a + b) = (a + b) (a- ab + b^) 8. (a - b) = (a - b) (a² + ab + b2) 9. (a + b + c)2 = a + b + c2+2 (ab + bc + ca) ap + b3 + c3 - 3abc = (a + b + c)(a + b 2 + c - ab-bc - ca) If a+b+c= 0, then a + b3 + c3 = 3abc + - 10. OBJECTIVE QUESTIONS BASED ON THESE FORMULAE 1. .. + 1 1 If Vx+ =5, find the value of x2 + Tx x2 (1) 527 (3) 427 (4) 327 (2) 224 ...
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