Polynomial evaluation ($E_a(p)=p(a)$)

Polynomial evaluation replaces the indeterminate of a polynomial by an input value. For a fixed scalar $a$, the rule $E_a:p\mapsto p(a)$ is a linear functional on a vector space of polynomials, because evaluating a linear combination gives the same linear combination of the values. For $p(t)=1+2t+3t^2$, evaluation at zero gives one and evaluation at two gives seventeen. Polynomial interpolation uses these measurements to reconstruct a polynomial subject to a polynomial degree bound.

The residual of a proposed polynomial root $\hat{z}$ is the value $p(\hat{z})$, which is zero for an exact root. Its size alone does not bound the root error without information about the scaling and derivative of the polynomial.