mportant Formulas in Algebra

Important Formulas in Algebra

Here is a list of Algebraic formulas –

• a² – b² = (a – b)(a + b)
• (a+b)² = a² + 2ab + b²
• a² + b² = (a – b)² + 2ab
• (a – b)² = a² – 2ab + b²
• (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc
• (a – b – c)² = a² + b² + c² – 2ab – 2ac + 2bc
• (a + b)³ = a³ + 3a²b + 3ab² + b³ ; (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴)
• (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴)
• a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
• a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)
• If n is a natural number, aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b+…+ b^n-2a + b^n-1)
• If n is even (n = 2k), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +…+ b^n-2a – b^n-1)
• If n is odd (n = 2k + 1), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +…- b^n-2a + b^n-1)
• (a + b + c + …)² = a² + b² + c² + … + 2(ab + ac + bc + ….
• Laws of Exponents (a^m)(aⁿ) = a^m+n (ab)^m = a^mb^m (a^m)ⁿ = a^mn
• Fractional Exponents a⁰ = 1 aman=am−n am = 1a−m a−m = 1am

• Roots of Quadratic Equation

• For a quadratic equation ax2 + bx + c where a ≠ 0, the roots will be given by the equation as −b±b2−4ac√2a
• Δ = b2 − 4ac is called the discrimination.
• For real and distinct roots, Δ > 0
• For real and coincident roots, Δ = 0
• For non-real roots, Δ < 0
• If α and β are the two roots of the equation ax2 + bx + c then, α + β = (-b / a) and α × β = (c / a).
• If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

• Factorials

• n! = (1).(2).(3)…..(n − 1).n
• n! = n(n − 1)! = n(n − 1)(n − 2)! = ….
• 0! = 1
• (a+b)n=an+nan−1b+n(n−1)2!an−2b2+n(n−1)(n−2)3!an−3b3+….+bn,where,n>1