8.4 For any field F, F[x],
c) P(x)Q(x) = P(x)R(x) [P(x) d.n.e. 0] implies that Q(x)=R(x).
My professor says that this is an induction problem. So I'm assuming that this is going to involve stating a particular term in LHS is equal to the same term in the RHS, up until a certain term. When multiplying two polynomials, let and represent the sum of all coefficients of the term when the two polynomials are multiplied. Basically just grouping all like powers and having one coefficient per power. Let's say E(k) is a propositional statement that " ", where is the coefficient of the respective term in the product of P(x)Q(x) and is the coefficient in the product of P(x)R(x).
So now assume that P(k) is true for all k<i. We must now show that P(i) -> P(i+1), then all terms of the two products are equal and Q(x)=R(x).
Look like a good plan?