An introduction to the modern theory of equations, by Florian Cajori.

204

THElORY OF EQUATIONS

Take 7 -- + The quadratic whose roots are w and w4 is
2
x2 - (0 + W4)x + W * W4 = 0,
or. 2 - + 1 = 0.
Whence x_ = -1 + 1 = v + 10 + 2Vl /5
2 Y 4 4
According to ~ 183 the inscription of a regular pentagon into a circle
can be effected with the aid of ruler and compasses.
Ex. 2. Solve x3 - 1 = 0.
Here n - 1 = 3. 2. 2. Hence the solution of one cubic and two quadratics is called for, and the inscription of a regular polygon of thirteen
sides into a circle by ruler and compasses is impossible. Take g 6, then
the roots of - 1= 0 are
a-i
CW, 0g, 0092,..., 0911,
or W,, co 10 o 9, 2 W, w212, w7, w3, w5, w4, W11.
If we taken - 1 = e f= 12 3. 4, where e 3, we get
n1 -W + - w8 + w12 + C5,
1- Er ' + w9 + W7 + 4,
772 10 + W2 W3 -+ W1.
To compute the cubic of which v, 71, 72 are roots, we obtain
77 + 771+, 7 -+ - -1,
7M71 = 2 7 + 1q + 772,
77112 = 77 + 2 771 - T2,
77772 = 7+ 771 + 2 772,
77 = 4 + 2 77i - )2,
77771772 = 77 + 2 77771 + 77 - 1,
7771 + 711w + 712 = 4(77 + 771 + 712)=- 4.
The cubic is X3 + x2 - 4 x + 1 = 0. Solving this, we obtain the values of
77, l, 772.
Take next 4 = e'lf = 2.2. We have 77' - w12, v'3 = s8 -+ 5.
Since 7' + 7713 -= and 77'7'3 - 71, we find that 77' and 7'3 are roots of the
quadratic - 0,
X2 - -nX +- 771 = 0,