No theorem of elementary algebra has given rise to more discussion than the Remainder Theorem:

If a polynomial f(x) be divided by (x — a), the remainder is f(a).

It is hoped that this note may be the means of settling questions connected with it once for all, and may aid in bringing about correct methods in teaching.

The following note suggests certain connections between the theory of linear transformations and quadratic forms on the one hand, and the geometry of second degree surfaces on the other. It is hoped that the note may prove useful to those who may have to teach either theory to students who already possess an elementary knowledge of the other. The general ideas may be such as may well have occurred to anyone familiar with both theories, but the examples given may be new to readers.

The condition that ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 should represent two straight lines can be obtained very easily by considering the product

which is equal to

and is identically zero.

The following proof is designed to link up Hero's formula geometrically with the formulae for the trigonometrical functions of ½A in a triangle.

The triangle ABC has a right angle at B. On AC, on the same side as B, describe a square ADEC. Draw DF perpendicular to AB or AB produced.

The triangles ABC and DFA are congruent, having sides CA and AD equal, and the corresponding angles equal. Hence DF is equal to AB.

1. In one of Sir Thomas Muir's more recent historical papers on the theory of determinants (Proc. R. S. Edin., XLIII, 1922, p. 129), is included a result due to V. Jung, commented on as being “verified in an unsuggestive way for the first three cases.” The theorem is that the determinant

Vanishes when n is even. Below we give a simple proof.

The six values of x which make the determinant

vanish, are . In general, the n values of x which make the corresponding determinant of order n vanish are given by . Each determinnant has a diagonal filled with x's, bordered by adjacent parallels where each element is unity; and all other elements are zero.