First Course in the Theory of Equations by Leonard Eugene Dickson

By Leonard Eugene Dickson

This can be a new printing of the vintage booklet by means of Dickson. It was once to satisfy the varied wishes of the scholar in regard to his prior and destiny mathematical classes that the current booklet was once deliberate with nice care and after broad session. It differs primarily from the author's "Elementary idea of Equations", either in regard to omissions and additions, and because it truly is addressed to more youthful scholars and should be used parallel with a direction in differential calculus. less complicated and extra specified proofs are actually hired. The routines are less complicated, extra various, of larger kind, and contain simpler purposes. There are greater than 480 workouts for college students to perform. subject matters lined contain complicated numbers, roots of equations, the quadratic equation, graphical equipment, cubic and quartic equations, numerical tools, platforms of linear equations, and so on. solutions to all of the routines are supplied on the finish of the publication. this can be a new, top of the range, and cheap version.

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With a common ratio r [say m/r, m, mr]. 9. , with a common difference d [say m − d, m, m + d]. 10. Solve x4 − 2x3 − 21x2 + 22x + 40 = 0, whose roots are in arithmetical progression. [Denote them by c − 3b, c − b, c + b, c + 3b, with the common difference 2b]. 11. Find a quadratic equation whose roots are the squares of the roots of x2 − px + q = 0. 22 THEOREMS ON ROOTS OF EQUATIONS [Ch. II 12. Find a quadratic equation whose roots are the cubes of the roots of x2 − px + q = 0. Hint: α3 + β 3 = (α + β)3 − 3αβ(α + β).

Hence there is no rational root. Hence (§§30, 31) it is not possible to duplicate a cube with ruler and compasses. 34. Regular Polygon of 7 Sides. If we could construct with ruler and compasses an angle B containing 360/7 degrees, we could so construct a line of length x = 2 cos B . Since 7B = 360◦ , cos 3B = cos 4B . But 2 cos 3B = 2(4 cos3 B − 3 cos B) = x3 − 3x, 2 cos 4B = 2(2 cos2 2B − 1) = 4(2 cos2 B − 1)2 − 2 = (x2 − 2)2 − 2. Hence 0 = x4 − 4x2 + 2 − (x3 − 3x) = (x − 2)(x3 + x2 − 2x − 1).

For example, 1 1 y4 + 4 = x y3 + 3 y y 1 − y2 + 2 y = x(x3 − 3x) − (x2 − 2) = x4 − 4x2 + 2. Or we may employ the explicit formula (19) of §107 for the sum y k + 1/y k of the kth powers of the roots y and 1/y of y 2 − xy + 1 = 0. ] 43 REGULAR POLYGON OF 9 SIDES 37. Regular Polygon of 9 Sides and Roots of Unity. If R = cos 2π 2π + i sin , 9 9 the powers R, R2 , R4 , R5 , R7 , R8 , are the primitive ninth roots of unity (§11). They are therefore the roots of y9 − 1 = y 6 + y 3 + 1 = 0. y3 − 1 (19) Dividing the terms of this reciprocal equation by y 3 and applying the second relation (15), we obtain our former cubic equation (11).

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