Question

INSANELY EASY question, lack of sleep, need answer

Little brother wants to know the difference between a numerical proof, and an algebraic proof, when solving a polynomial equation perhaps giving an example of each. I am running on less than an hour of sleep here and I don't want to give him a wrong answer due to my lack of sleep.

Answer 1

When I think of "solving" a polynomial equation, my goal is to find all values of the independent variable (usually x) for which the equation is true. If we can do this algebraically, not needing a calculator, that's an algebraic procedure (not proof).

Supposing that your polynomial is not so "nice," meaning that you need to approximate roots to so many decimal places. That's a numerical (methods) solution. Newton's Method is the primary example of that. The degree of accuracy you can obtain through this method is limited only by the number of decimal places your calculator or computer software can handle.

Answer 2

turning this over to my little brother here-
would factoring a polynomial equation be doing it algebraicly?

Answer 3

yes :)

Answer 4

and solving through simplification would be numerical right?

Answer 5

No, at least not in my way of thinking. When I see the word "numerical" in this context, I think "numerical methods," which includes Newton's Method and various others.

Answer 6

okay, thanks!