See the entire solution process below: Explanation: First, remove all of the terms from parenthesis. Be careful to handle the signs of each individual term correctly: \displaystyle{3}{x}^{{2}}+{4}{x}-{10}-{2}{x}-{7}+{4}{x}^{{2}} ...

Zeros: \displaystyle-{1},{3},-{4},-{5} Explanation: \displaystyle{f{{\left({x}\right)}}}={x}^{{4}}+{7}{x}^{{3}}-{x}^{{2}}-{67}{x}-{60} By the rational root theorem, any rational zeros of \displaystyle{f{{\left({x}\right)}}} ...

You have forgotten that the x y^2 term requires a product rule when differentiating

Basic definitions Before you can understand finite fields, you need to understand what a field is. Fields are algebraic structures, meant to generalize things like the real or rational numbers, ...

An integral domain D embeds in its field of fractions F say, so a polynomial can't have more roots in D than it does in F. If you don't require multiplication to be commutative or if you allow ...

You've shown that f(a)\equiv0\pmod{2} for a=1 and a=2 . To show that there exists some \alpha\in\Bbb{Z}_p such that f(\alpha)=0 , you can apply Hensel's lemma as you quote it in your ...