Here we work in the system of integer polynomials. Those are polynomials of the Form f(x)=rnxn+···+r1x+r0 where every coefficient is an integer.
General question:
When does some combination of the polynomials ax + b and cx + d equal 1 ? That is, when do there exist integer polynomials P(x) and Q(x) with
P(x)·(ax + b) + Q(x)·(cx + d) = 1 ? We concentrate here on cases when c = 0.
(a) Prove: No combination of 2x + 5 and 3 can equal 1. That is, no integer polynomials P (x), Q(x) can satisfy:
P (x)·2x + 5 + Q(x)·3 = 1.
(b) Find a combination of 2x + 5 and 4 that equals 1.
(c) Does some combination of 15x+9 and 25 equal 1? How about 15x+9 and 20? Explain your reasoning.
(d) Investigate further examples of ax + b and d, deciding in each case whether 1 is a combination. What patterns do you detect?
Can you prove that some of your observed patterns always hold true?