Algebra Formulas

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 + 2bc + 2ca
• (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca
• (a + b)³ = a³ + 3a²b + 3ab² + b³ ; (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – 3a²b + 3ab² – b³ = a³ – b³ – 3ab(a – b)
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)
• (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 +a^n-3b²…- 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 ; $\frac{a^{m}}{a^{n}} = a^{m-n}$ ; $a^{m}$ = $\frac{1}{a^{-m}}$ ; $a^{-m}$ = $\frac{1}{a^{m}}$

• Roots of Quadratic Equation

• • For a quadratic equation ax² + bx + c = 0 where a ≠ 0, the roots will be given by the equation as \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)
  • Δ = b² − 4ac is called the discriminant
  • 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 = 0 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
  • \((a + b)^{n} = a^{n}+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^{2}+\frac{n(n-1)(n-2)}{3!}a^{n-3}b^{3}+….+b^{n}, where\;,n>1\)

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