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76 2. Evaluation, Division, and Expansion

2. PI, P2, P3 are quadratic polynomials with positive leading coefficients
   and real zeros. Show that, if each pair of them has a common zero,
   then the trinomial 4 + P2 + P3 also has real zeros.


3. Show that, if n is a positive integer greater than 1,

--

r:, - (:$2

is a polynomial in x of degree n - 2, and find its coefficients

(a) when it is arranged in powers of x,

(b) when it is arranged in powers of (x - 1).

4. If n is a positive integer, prove that

(1-x)3”+3n+(l-x)3”-2+ 3n’:y, 3)x2(1-x)3”-4+. -. = (l-x3)“.

5. Suppose that ac - b2 # 0. Consider the equation

ax3 + 3bx2 + 3cx + d = 0.

(a) Show that the equation has two equal roots if and only if

(bc - ad)’ = 4(ac - b’)(bd - c”).

(b) Show that, if the equation has two equal roots, they are each

equal to

6. Prove that

(bc - ad)/[2(ac - b’)].

(n + l)‘+l -(n+l)=(r+l)S,+ ‘11 S,-i+...+(r+l)Si

( >

where S, = 1’+2’+3’+...+n’.

7. Find all polynomials p of degree k with real coefficients for which
   p(p(t)) is a positive integer power of p(t).


8. Find all polynomials p and q for which p(t) = q(p’(t)).


9. (a) Find all polynomials p(t) of degree not exceeding 3 which com-
   mute with their first derivatives, i.e. for which

P(P’W) = p’(p(t)).

(b) For each integer n > 4, determine a polynomial of degree n

which commutes with its derivative.