Relationships between the roots and coefficients of a quartic equation

Relationships Between Roots and Coefficients of a Quartic Equation

Quartic equations are polynomial equations of the fourth degree. A general quartic equation can be written as: ax⁴ + bx³ + cx² + dx + e = 0 where a ≠ 0.

A quartic equation with real coefficients has either 0, 2 or 4 real roots. Remaining roots, if any, are complex and exist in conjugate pairs.
The sum of the roots of the quartic equation is -b/a. This is called Viète’s first formula.
The sum of the products of all possible pairs of roots is c/a. This corresponds to Viète’s second formula.
The sum of products of all possible triplets of roots results in -d/a. This is another relation according to Viète’s formulas.
The product of all the roots is given by e/a. If ‘a’ is positive, the sign of this product will depend on the number of negative real roots. This equation represents Viète’s fourth formula.

These relationships are useful in solving quartic equations, particularly when the roots are complex or inconvenient to find.
Understanding the relationships simplifies the calculation in scenarios where only certain characteristics of the roots are required, without needing to find the roots themselves.
These formulas are also applicable in solving equations in other disciplines such as physics and economics that employ polynomial equations.

The root-coefficients relationships can be derived by expanding (x - α)(x - β)(x - γ)(x - δ), the factored form of the quartic equation, where α, β, γ and δ represent the roots.