2.5. Problems 77
- Let zcn) = z(z - 1)... (z - n + 1) for n a positive integer and let
t(O) = 1. Prove that
(x + y)(n) = 2 ( ; ) .(k)y(“-k).
k=O
- Determine a polynomial solution of the differential equation
203/1’+9~+4y’+y=x3+5x2-2X-2,
where y is to be found as a function of x.
- Find a polynomial j(x) of degree 5 such that f(x) - 1 is divisible by
(x - 1)3 and f(x) is itself divisible by x3. - If the polynomial asx3 + a2x2 + arx + as (as # 0) is the third power
of a linear polynomial, prove that
9aoa3 = ala2
and
Prove the converse: if these two conditions are satisfied, then the
polynomial is the third power of a linear polynomial.
- Let k be the smallest positive integer with the property:
There are distinct integers a, b, c, d, e such that p(x) =
(X - u)(x - b)(x - c)(x - d)(x - e) has exactly k nonzero
coefficients.
Find with proof, a set of integers a, b, c, d, e for which the minimum
is achieved.
- Define polynomials fn(x) for n = 0, 1,2,.. ., by
lo(x) = 1
ha(O) = cl (n 2 1)
fL+&) = (n + l>fn(x + 1) (fl^2 0).
Find, with proof, the explicit factorization of free(l) into powers of
distinct primes.
- Find polynomials f(x) such that
f(x”) + f(x)f(x + 1) = 0.