Two polynomials are considered to be equal if and only if the corresponding coefficients for each power of X are equal.

The topology on R'X is such that a sequence of its elements converges only if for each monomial X the corresponding coefficient stabilizes.

Large values of t indicate that the null hypothesis can be rejected and that the corresponding coefficient is not zero.

The fact that the hyperbolic plane has an infinite area comes by computing the surface integral with the corresponding coefficients of the First fundamental form.

Moreover, once SMB and HML are defined, the corresponding coefficients and are determined by linear regressions and can take negative values as well as positive values.

This argument shows that more generally, any irreducible factor of P can be supposed to have integer coefficients, and leading and constant coefficients dividing the corresponding coefficients of P.

These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist).

Addition and subtraction are performed by adding or subtracting corresponding coefficients.

Multiplication is performed much the same way as addition and subtraction, but instead by multiplying the corresponding coefficients.

It relies on the fact that two polynomials are identical precisely when all corresponding coefficients are equal.