Polynomial Equations
Definition: Polynomial Equation
A polynomial equation over a field \(F\) is an Equation of the form
\[ P(x_1, \cdots, x_n) = 0_F, \]
where \(P(x_1, \cdots, x_n)\) is a nonzero polynomial.
Definition: Solution of a Polynomial Equation
A solution to the polynomial equation \(P(x_1, \cdots, x_n) = 0_F\) is an \(n\)-Tuples \(S = (\lambda_1, \cdots, \lambda_n)\) of elements from \(F\) such that the value of \(P\) at \(S\) is \(0_F\).
Polynomial Equation in a Single Variable
A polynomial equation in a single variable has the form
\[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0 \]
Definition: Quadratic Equation
A quadratic equation is a polynomial equation \(P = 0\), where \(P\) is a [[Quadratic Polynomials|quadratic polynomial]].
Quadratic Equation in a Single Variable
A quadratic equation in a single variable has the form
\[ a x^2 + bx + c = 0 \]