21. Prove that if r₁, r₂..., r_n are distinct roots of a polynomial equation f(x) = 0, then f(x) is a multiple of (x - r₁)(x - r₂)...(x - r_n).

22. Prove thatare all irrational.

23. Prove thatare all irrational.

24. Prove thatis irrational.

25. Find an eighth-degree polynomial equation with integer coefficients that has as a root.
 

26. If f(x) is a function of x, the notationrepresents f(x + 1) - f(x). Show that 
 

27. Let  Find  for each of the following:
 

(a) f(x) = a + bx.
 

(b) f(x) = a + bx + cx².
 

(c) f(x) = a + bx + cx² + dx³.
 

(d) f(x) = xⁿ, with n a positive integer.
 

28. Find f(x + 2) - 2f(x + 1) + f(x) for:
 

(a) f(x) = a + bx.
 

(b) f(x) = a + bx + cx².
 

29. Find f(x + 3) - 3f(x + 2) + 3f(x + 1) - f(x) for:
 

(a) f(x) = a + bx.
 

(b) f(x) = a + bx + cx².
 

(c) f(x) = a + bx + cx² + dx³.

30. Let with n a positive integer, be defined inductively by

[The functionis called the nth difference of f(x).] Show that

Monday, June 15, 1998