Unknown

2.5. Problems 77

10. 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

11. 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.

12. 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.


13. 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.

14. 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.

15. 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.

16. Find polynomials f(x) such that

f(x”) + f(x)f(x + 1) = 0.