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Hints 45

1.20. p(t2) has terms in only even powers of t.

2.3. One root is t = m.

2.5. The discriminant should vanish.

2.8. The equation leads to a quadratic in x. For which values of Ic is the

discriminant nonnegative?

2.10. Express the sum and product of m - n2 and n - m2 in terms of

m + n = 5/6 and mn = -l/2.

2.14. (a) The polynomial does not take negative values.

2.17. The discriminant of the quadratic, i.e. (fi+ &)2 -4fi, is not less

than 0.

3.3. (k) For a clean proof, apply (h), (d) and (f) to 1% + ~1~.

3.5. The given locus is l/w times the locus of Re(z) = c.

3.7. Let U and V be represented by the points 0 and 1 in the complex

plane, and suppose the tree T is at z. Locate the points P and Q,

noting that multiplication by i corresponds to a rotation through a

right angle. Show that the midpoint of PQ does not depend on z.

3.10. The solution of the equations for x and y in terms of a and b can be
facilitated using the theory of the quadratic.

3.14. (a) Make use of Exercise 3(c) and 3(d).

3.15. (d) Use de Moivre’s Theorem, Exercise 3.8.

4.4. (b) Use Exercise 2.4.

4.9. To convert the particular equation to the general form, let y = 2x.

4.16. (b) Let u be the polynomial g with its coefficients in the opposite
order, i.e. u(t) = tkg(l/t) where JC = degg. Show that uh = f = gh
and use Exercise 1.17.

5.5. It suffices to prove the result for polynomials of the form x”y’ + xbya.

5.7. Three numbers are in arithmetic progression if and only if their sum

is equal to three times one of the numbers.

5.9. (c) Let Q = x3, etc., and apply (a) and (b).

6.3. (c) Multiply the equation in (b) by b.

6.5. (b) Note that ac - bd = a(c - d) + (u - b)d.