1.1. The Anatomy of a Polynomial of a Single Variable 5
(4 degtp + d 6 max(deg p, deg q). ( max(a, 6) is the larger of the
two numbers a and b)
(b) de&q) = deep + deg q.
Give examples when equality and strict inequality hold in (a). Ob-
serve that, because of the convention that the sum of --oo and any
nonnegative number is -00, the degree of the zero polynomial is de-
fined in such a way as to make (b) valid when one of the polynomials
is zero.- Is deg(p o q) related in any way to deg(q op)?
- Find a pair p, q of polynomials for which p o q = q o p,
- (a) Is it possible to find a polynomial, apart from the constant 0
itself, which is identically equal to 0 (i.e. a polynomial p(t) with
some nonzero coefficient such that p(c) = 0 for each number
c)? Try to justify your answer. [This is not an easy question,
although the answer is not surprising. Examine your justification
carefully to see what you are assuming about polynomials; can
you explain why it is valid to use the properties you think you
need?]
(b) Use your answer to (a) to deduce that, if two polynomials assume
exactly the same values for all values of the variable, then their
respective coefficients are equal. [Thus, there is only one way, up
to order of writing down the terms, of presenting a polynomial
as a sum of monomials.] - (a) Find all polynomials f such that f(2t) can be written as a poly-
nomial in f(t), i.e. for which there exists a polynomial h such
that
f(W = W(t)).
(b) Use the identity sin2 2t = 4sin2 t(1 - sin2 t) to show that sint is
not a polynomial. - Show that, for t > 0, log t is not a polynomial.
- Find all periodic polynomials, i.e. polynomials g(t) which satisfy an
identity of the type g(t + Jc) = g(t) for some k and all t. Deduce that
the trigonometric functions sin t, cost and tan t are not polynomials. - Prove that tâi3 is not a polynomial.
- Show that, if p, f, g are nonzero polynomials for which pf = pg, then
f=s*