By George Chrystal

ISBN-10: 1402151535

ISBN-13: 9781402151538

This Elibron Classics booklet is a facsimile reprint of a 1904 variation by means of Adam and Charles Black, London.

**Read or Download Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 PDF**

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**Extra info for Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 **

**Example text**

E. if the A A A A division ratios * 3 and * 4 are opposite numbers, we say (AZÌAQ) ■"2-^-3 A2A^ that the pairs (Al9 A2) and (AZ9 AA) separate each other harmo nically or that the quadruple of points Al9 A29 AZ9 A^ is harmonic. This case is shown in Fig. 6, in which x1 , A, 0 rH 0 , x3 I I A3 x2 *4 I A2 A4 FIG. 6 A2AZ A2A^ 39. Method I. We know from algebra that the numbers xl9 x2, xz are the roots of the equation x3+mx2+nx+p = 0 if and only if the following conditions are satisfied : *^1 *^2~Γ~^2^3 i •^S^'i == Wi x±x2xz = — p.

A*+a*+... +x) +l = χ«(χ*+χ3+... + 1)+χ{χ*+χ*+... + 1) + 1 = (x*+x3+... + l)(x«+x) + l. Formula (4) can therefore be written in the form (x*+x*+... + l)(x«+x)(x**+x™+... + l)+(x**+x™^ = (x*+x*+... + l)(x50+x*5+... + l). Let us subtract the first term of the equality from each side, and then let us factorize the right-hand side ; we obtain z 4 4 +z 3 3 +... + l = ( ^ + x 3 + . . + l)[a; 50 +ir 45 +... + l ~ -(x«+x)(x**+x**+... + l)]. (5) 11 Equality (5) shows that the polynomial a^+x^+x^+x -^!

6) 2 We substitute the expression for x from formula (6) in equation (5): —(m—p)(mx+n) + (n—q)x = 0. Hence [—m(m— p) + (n—q)]x—n(m— p) = 0. (7) Now we eliminate x from the linear equations (3) and (7). Accordingly we multiply equation (3) by [—m(m— p)+{n—q)] and equation (7) by — (m—p) and obtain by addition: (n—q)[—m(m— p) + (n—q)]+n(m— p)2 = 0. Hence (n—q)2—(m—p)[m(n—q)—n(m—p)] = 0 and finally (n—q)2—(m—p)(np—mq) = 0. Method II. If one of the roots xx, x% of the first equation of (1) is equal to one of the roots of the second equation, then one of the differences xx—xZ) x%—xZi xx—xé, χ^—χ^ is equal to zero.

### Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 by George Chrystal

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