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320 Answers to Exercises and Solutions to Problems

The roots of x3 +px2 + qx + r = 0 are vertices of an equilateral triangle iff

they have the form -p/3 + u, -p/3 + uw, -p/3 + uw2 for some nonzero u.

This happens iff the cubic equation can be written in the form (x+P/~)~ =

u3 for some nonzero u, which occurs iff p2 = 3q, r = p3/27 - u3, i.e.

p” # 27r.

A generalization can be found in Crux Mathematicorum 9 (1983), 218-

221.

9.4. First, we determine the area of a triangle with vertices u, v, w in the

complex plane. This area remains unchanged if we subject the points to

a translation (subtraction of u) followed by a rotation (multiplication by

Iv-~I-~(iT--zi)) t o o bt ain the vertices 0, Iv - u], Iv - u]-~(w - u)(‘ii- 21).

Using the half-base-times-height formula for area yields

$1~ - ul. Im[]v - ul-‘(w - u)(is - G)]

Refer to Exercises 1.4.2-4. A change of variables x = y+a/3 converts the

equation to one of the form y3 + py + q = 0 whose zeros are obtained from

those of the given equation by a translation and whose coefficients p, q are

expressible in terms of a, b, c. If u, v, w are the zeros of the polynomial in y,

then w = -(u + v) and the formula for the area reduces to (3/4)]u’ii - UV].

Now u = r+s, v = rw+sw2, where 18r3 = -9q+dm, s = -p/3r

and w is an imaginary cube root of unity. The area is thus

lw2 - wl(lr12 - ls12) = di(IrI” - ls12).

9.5. If the zeros are u, iv, -iv (u, v real), then b = -au, c = uv2 and

d = -auv2, from which the necessity of the conditions follow.

On the other hand, if UC > 0 and bc = ad, we have that

at3 + bt2 + ct + d = (at + b)(t2 + a-‘~),

from which the sufficiency of the conditions can be deduced.

9.6. Taking note of the fact that ]a] = Ibl = ICI = IdI = 1, we find that,

when ]z] = 1,

Ip( = p(z)p(z> = 4 + (ud)z3 + (Ed)z-3 + (U-E + bz)z2

+ (tic + 6d)z-” + (a8 + bE + cz)z + (Eb + bc + Ed)%-‘.

Let w be an imaginary cube root of unity. Since 1 + w + w2 = 0,

Ip( + Ip(wz)l” + lp(w2z)12 = 12 + 3~2%~ + 3sidz-3.

Choosing t to be a cube root of zd gives the right side the value 18, whence
at least one of the three terms on the left side is at least 6. The result follows.