76 2. Evaluation, Division, and Expansion
- 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. - Show that, if n is a positive integer greater than 1,
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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).
- 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)“.
- 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
- 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’.
- Find all polynomials p of degree k with real coefficients for which
p(p(t)) is a positive integer power of p(t). - Find all polynomials p and q for which p(t) = q(p’(t)).
- (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.