Algebra

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Important Maths Formulas of 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 =Formula_exponent
  • 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:Quadratic Formula
  • If D = 0, the quadratic equation has repeated real roots given by:

Quadratic Equation equal roots

  •  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.:
    Sum of n terms of an AP
  • Here, n is the total number of terms

        last term of an APis the last term

      an is the nth term

Relationship between Coefficients and Zeroes of a Polynomial
  • Linear:
    p(x) = ax + b , a ≠ 0; xα

Relation between zero and coefficients of a linear polynomial

  • Quadratic:
    p(x) = ax2 + bx + c, a ≠ 0; xαβ Formula_Roots and coefficient relationCubic:
    p(x) = ax3 + bx2 + cx + d, a ≠ 0; xαβ,γ
    Relation between zero and coefficients of a cubic polynomial1
    Relation between zero and coefficients of a cubic polynomial2Relation between zero and coefficients of a cubic polynomial 3
Relationship between HCF and LCM of two numbers

 

For any two positive integers a and b,

  •  LCM (a, b) × HCF (a, b) = a × b