Theory of Equations
Theory of Equations tests the relation between roots and coefficients, formation of equations, the nature of roots, and transformations of a polynomial equation.
Vieta's formulas
For ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = −b/a, Σαβ = c/a, αβγ = −d/a. For ax² + bx + c = 0: α + β = −b/a and αβ = c/a. A polynomial of degree n has exactly n roots counted with multiplicity, and a polynomial with real coefficients has complex roots in conjugate pairs.
Derived quantities
Σα² = (Σα)² − 2Σαβ; Σ1/α = Σαβ/αβγ; Σα²β² = (Σαβ)² − 2αβγΣα. To form an equation from given roots: x³ − (Σα)x² + (Σαβ)x − αβγ = 0.
Transformations
Reciprocal roots: replace x by 1/x. Roots multiplied by k: replace x by x/k. Roots diminished by h: replace x by x + h. Symmetric reciprocal equations are solved by dividing by x² and substituting u = x + 1/x.
Worked example
If α, β, γ are the roots of x³ − 3x² + 2x − 1 = 0, then Σα = 3, Σαβ = 2 and αβγ = 1. Hence Σα² = 3² − 2(2) = 5 and Σ1/α = 2/1 = 2. No individual root has to be found.
Traps
Reversing the alternating signs; using αβγ = d/a; assuming integer roots because the coefficients are integers; applying a transformation and then forgetting to rename the variable back.