# Algebra

 Algebraic Identities (a + b)2 = a2 + 2ab + b2 a2 + b2 = (a + b)2 − 2ab (a − b)2 = a2 − 2ab + b2 a2 + b2 = (a − b)2 + 2ab a2 − b2 = (a + b)(a − b) (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) (a + b)3 = a3 + b3 + 3ab(a + b) a3 + b3 = (a + b)3 − 3ab(a + b) (a − b)3 = a3 − b3 − 3ab(a − b) a3 − b3 = (a − b)3 + 3ab(a − b) a3 − b3 = (a − b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 − ab + b2) a3 + b3 + c3 - 3abc =(a + b +c )(a2 +b2 +c2 – ab – bc – ac) If a + b + c = 0, then a3 + b3 + c3 = 3abc Exponents an = a.a.a . . . n times am. an = a m+n  am/ an = am−n if m > n = 1 if m = n = 1/ an−m if m< n; a ∈ R; a ≠ 0 (am)n = amn = (an)m (ab)n = an.bn a0 = 1 where a ∈ R; a ≠0 a−n = 1/an ; an = 1/a−n a p/q = If am = an and a ≠ ± 1; a ≠0  then m = n  If an = bn where n ≠ 0, then a  = ± b Quadratic Equations Ø  For a quadratic equation: ax2 + bx + c = 0,  D = b2 – 4ac If D > 0, the quadratic equation has real and distinct roots given by the quadratic formula: If D = 0, the quadratic equation has repeated real roots given by: If D < 0, the quadratic equation does not have real roots. Ø  The quadratic equation whose roots are α and β is: (x − α)(x − β)=0 i.e. x2 − (α + β)x + αβ = 0 i.e. x2 − Sx + P = 0 where S = Sum of the roots and P = Product of the roots Arithmetic Progression Let a be the first term and the d be the common difference of an A.P, then: General form: of an A.P.: a, (a + d), (a + 2d), (a + 3d),… Finding the nth term of an A.P:  an = a + (n - 1)d Sum of n terms of an A.P.: Here, n is the total number of terms is the last term       an is the nth term Relationship between Coefficients and Zeroes of a Polynomial Linear: p(x) = ax + b , a ≠ 0; x = α Quadratic: p(x) = ax2 + bx + c, a ≠ 0; x = α, β Cubic: p(x) = ax3 + bx2 + cx + d, a ≠ 0; x = α, β,γ   Relationship between HCF and LCM of two numbers For any two positive integers a and b,  LCM (a, b) × HCF (a, b) = a × b