Unknown

1.4. Equations of Low Degree 19

(b) Show that u3 and v3 are roots of the quadratic equation

x2 + qx - p3/27 = 0.

(c) Let D = 27q2 + 4p3. Suppose that p and q are both real and

that D > 0. Show that the quadratic in (b) has real solutions,

and that if us and us are the real cubic roots of these solutions,

then the system in (a) is satisfied by

(u, v) = (uo, vo), (uow, vow2), (uow2, VW)

where w is the imaginary cube root (-1 + -)/2 of unity.

Deduce that the cubic polynomial t3 +‘pt + q has one real and

two nonreal zeros.

(d) Suppose that p and q are both real and that D = 0. Let uc be

the real cube root of the solution of the quadratic in (b). Show

that, in this case, the cubic has all its zeros real, and in fact can

be written in the form

(t + “o)2(t - 2uo).

(e) Suppose that p and q are both real and that D < 0. Show

that the solutions of the quadratic equation in (b) are nonreal

complex conjugates, and that it is possible to choose cube roots u

and v of these solutions which are complex conjugates and satisfy

the system in (a). If u = P(COS B+i sin 6) and v = r(cos 0-i sin e),

show that the three roots of the cubic equation are the reals

2r cos 8,2r cos(fl + 2~/3), 2r cos(0 + 4~/3).

(f) Prove that every cubic equation with real coefficients has at least

one real root.

5. Use Cardan’s Method to solve the cubic equations:

(a) x3 - 6x + 9 = 0.

(b) x3 - 7x + 6 = 0.

[(b) will require the use of a pocket calculator and some trigonometry;

remember de Moivre’s Theorem (Exercise 3.8); work to an accuracy

of three decimal places.]

6. By means of a transformation, convert the equation

x3 - 15x2 - 33x + 847 = 0

to the form t3 +pt + q = 0, and verify that D = 0 (in the notation of

Exercise 4). Solve the given equation for x.