roots of polynomials

• Roots of a quadratic equation:

  • A quadratic equation could have 2 real roots or 2 complex roots
  • If 𝜶 and 𝜷 are roots of the equation 𝒂𝑥² + 𝒃𝑥 + 𝒄 = 0, then:
  • 𝜶 + 𝜷 = -𝒃/𝒂
  • 𝜶𝜷 = 𝒄/𝒂
• Roots of a cubic equation:

  • A cubic equation could have 3 real roots or 1 real root and 2 complex roots
  • If 𝜶, 𝜷, and 𝜸 are roots of the equation 𝒂𝑥³ + 𝒃𝑥² + 𝒄�� + 𝒅 = 0, then:
  • 𝜶 + 𝜷 + 𝜸 = -𝒃/𝒂
  • 𝜶𝜷 + 𝜷𝜸 + 𝜸𝜶 = 𝒄/𝒂
  • 𝜶𝜷𝜸 = -𝒅/𝒂
• Roots of a quartic equation:

  • A quartic equation could have 4 real roots, 4 complex roots, or 2 real and 2 complex roots
  • If 𝜶, 𝜷, 𝜸, and 𝜹 are roots of the equation 𝒂𝑥⁴ + 𝒃𝑥³ + 𝒄𝑥² + 𝒅𝑥 + 𝒆 = 0, then:
  • 𝜶 + 𝜷 + 𝜸 + �� = -𝒃/𝒂
  • 𝜶𝜷 + 𝜶𝜸 + 𝜶𝜹 + 𝜷𝜸 + 𝜷𝜹 + 𝜸𝜹 = 𝒄/𝒂
  • 𝜶𝜷𝜸 + 𝜶𝜷𝜹 + 𝜶𝜸𝜹 + 𝜷𝜸𝜹 = -𝒅/𝒂
  • 𝜶𝜷𝜸𝜹 = 𝒆/𝒂
• • 𝜶³ + ��³ + 𝜸³ = (�� + �� + ��)³ - 3(𝜶 + 𝜷 + 𝜸)(���� + 𝜷𝜸 + 𝜸𝜶) + 3𝜶𝜷𝜸
• Expressions relating to the roots of a polynomial:

  • Reciprocals:
  • 1/𝜶 + 1/𝜷 = (𝜶 + 𝜷)/𝜶𝜷
  • 1/𝜶 + 1/𝜷 + 1/𝜸 = (𝜶𝜷 + 𝜷𝜸 + 𝜸𝜶)/𝜶𝜷𝜸
  • 1/𝜶 + 1/𝜷 + 1/𝜸 + 1/𝜹 = (𝜶𝜷𝜸 + 𝜷𝜸𝜹 + 𝜸𝜹𝜶 + 𝜹𝜶𝜷)/𝜶𝜷𝜸𝜹
  • Products of powers:
  • 𝜶ⁿ × 𝜷ⁿ = (𝜶𝜷)ⁿ
  • 𝜶ⁿ × 𝜷ⁿ × 𝜸ⁿ = (𝜶𝜷𝜸)ⁿ
  • 𝜶ⁿ × 𝜷ⁿ × 𝜸ⁿ × 𝜹ⁿ = (𝜶𝜷𝜸𝜹)ⁿ
  • Rules for sums of squares:
  • 𝜶² + 𝜷² = (𝜶 + 𝜷)² - 2𝜶𝜷
  • 𝜶² + 𝜷² + 𝜸² = (𝜶 + 𝜷 + 𝜸)² - 2(𝜶𝜷 + 𝜷𝜸 + 𝜸𝜶)
  • 𝜶² + 𝜷² + 𝜸² + 𝜹² = (𝜶 + 𝜷 + 𝜸 + 𝜹)² - 2(𝜶𝜷 + 𝜶𝜸 + 𝜶𝜹 + 𝜷𝜸 + 𝜷𝜹 + 𝜸𝜹)
  • Rules for sums of cubes:
  • 𝜶³ + 𝜷³ = (𝜶 + 𝜷)³ - 3𝜶𝜷(𝜶 + 𝜷)
• Linear transformations of roots:

  • Given a polynomial, you can find the equation of a second polynomial whose roots are a linear transformation of the roots of the first
  • If a polynomial 𝑓(𝑥) = 𝑎𝑥⁴ + 𝑏𝑥³ + 𝑐𝑥² + 𝑑𝑥 + 𝑒 has roots 𝛼, 𝛽, 𝛾, and 𝛿, then the polynomial with roots 𝑔𝛼 + ℎ, 𝑔𝛽 + ℎ, 𝑔𝛾 + ℎ, and 𝑔𝛿 + ℎ, where 𝑔 and ℎ are real constants, is given by 𝑓(𝑤 - ℎ𝑔)
• k = ±ξ2
• m = -4/3k
• If k = ξ2, then m = -2ξ2/3
• If k = -ξ2, then m = 2ξ2/3
• α + β + γ = 4/7
• αβγ = -6/7
• α3β3γ3 = -216/343
• 1/α + 1/β + 1/γ = 1/6
• ∑α = 0
• ∑αβ = 2
• ∑αβγ = 1
• ∑αβγδ = 3
• 1/α + 1/β + 1/γ + 1/δ = 1/3
• α2 + β2 + γ2 + δ2 = -4
• (2α), (2β), (2γ), (2δ) are roots of the equation w4 + 4w3 - 10w2 + 8w - 8 = 0
• (α - 1), (β - 1), (γ - 1), (δ - 1) are roots of the equation 2w4 + 12w3 + 19w2 + 12w + 2 = 0