Chapter 4 - Roots of Polynomials

Quadratic roots (ax^2 + bx + c = 0): sum of roots

α + β = -b/a

Quadratic roots (ax^2 + bx + c = 0): product of roots

αβ = c/a

Cubic roots (ax^3 + bx^2 + cx + d = 0): sum of roots

α + β + γ = -b/a

Cubic roots (ax^3 + bx^2 + cx + d = 0): sum of pairwise products

αβ + βγ + γα = c/a

Cubic roots (ax^3 + bx^2 + cx + d = 0): product of roots

αβγ = -d/a

Quartic roots (ax^4 + bx^3 + cx^2 + dx + e = 0): sum of roots

α + β + γ + δ = -b/a

Quartic roots (ax^4 + bx^3 + cx^2 + dx + e = 0): sum of pairwise products

αβ + αγ + αδ + βγ + βδ + γδ = c/a

Quartic roots (ax^4 + bx^3 + cx^2 + dx + e = 0): sum of triple products

αβγ + αβδ + αγδ + βγδ = -d/a

Quartic roots (ax^4 + bx^3 + cx^2 + dx + e = 0): product of roots

αβγδ = e/a

Reciprocals (quadratic)

1/α + 1/β = (α + β)/(αβ)

Reciprocals (cubic)

1/α + 1/β + 1/γ = (αβ + βγ + γα)/(αβγ)

Reciprocals (quartic)

1/α + 1/β + 1/γ + 1/δ = (αβγ + βγδ + γδα + δαβ)/(αβγδ)

Product of powers (quadratic)

α^n × β^n = (αβ)^n

Product of powers (cubic)

α^n × β^n × γ^n = (αβγ)^n

Product of powers (quartic)

α^n × β^n × γ^n × δ^n = (αβγδ)^n

Sum of squares (quadratic)

α^2 + β^2 = (α + β)^2 − 2αβ

Sum of squares (cubic)

α^2 + β^2 + γ^2 = (α + β + γ)^2 − 2(αβ + βγ + γα)

Sum of squares (quartic)

α^2 + β^2 + γ^2 + δ^2 = (α + β + γ + δ)^2 − 2(αβ + αγ + αδ + βγ + βδ + γδ)

Sum of cubes (quadratic)

α^3 + β^3 = (α + β)^3 − 3αβ(α + β)

Sum of cubes (cubic)

α^3 + β^3 + γ^3 = (α + β + γ)^3 − 3(α + β + γ)(αβ + βγ + γα) + 3αβγ