8 Let Px be the predicate x is a car Let Qx be the predicate

8. Let P(x) be the predicate \"x is a car\". Let Q(x) be the predicate \"x is in the garage\". Let R(x) be the predicate \"x has a broken window\" Let S(x,y) be the predicate \"x and y are the same object\" a) Something is not in the garage. b) There are at least two cars. c) If something has a broken window and is in the garage then it is a car. d) Not everything has a broken window. e) There is exactly one car. f) Cars have broken windows. g) There is at most one car. h) Everything has a broken window and is in the garage. i) Only cars are in the garage Translate the following symbolic notation into English: j) Vx (R(x) Q(x)) k) Vx (Q(x) A R(x)) I) 3x (P(x) A Q(x)) m) 3x P(x) A 3x R(x)

Solution

the folloing propositions the

uppose p(x)p(x) is a polynomial with integer coefficients. Show that if p(a)=1p(a)=1 for some integer aathen p(x)p(x) has at most two integer roots.

Let p(x)p(x) be a cubic polynomial such that p(x)=ax3+bx2+cx+d=0p(x)=ax3+bx2+cx+d=0

Let p()=a3+b2+c+d=1p()=a3+b2+c+d=1 ( as per the question) .

where a,b,ca,b,c are integers.