Question

How do I prove this function is a polynomial?


Hi, I need to prove that (3y+2x)(5y−3x)=17y^2​​−7x^2 is a polynomial, using both an algebraic proof and a numerical proof. How do I do this?​

Answer 1

First and foremost, you'll need to inform yourself regarding the precise definition of a polynomial. Can you do that? What you have here is more complicated than the usual


\[y=ax^n+bx ^{n-1}+cx ^{n-2 }+ ... \]
format for a polynomial. You may have to decide for yourself whether to try solving for x or solving for y. If, for example, you chose to solve for y, you could try treating " x " as a constant.




Secondly, Multiply out the left side of the given equation

Lastly, you may need to apply the quadratic formula here to solve for n. If you do that, you'll likely end up with 2 solutions.

Answer 2

@mathmale I understand it's complicated; I had to write a polynomial identity for the assignment and I do agree I may have over complicated it...hopefully it still constitutes a polynomial, though?
Would you suggest I come up with something more simple before attempting to solve or write the required proofs for it?

Answer 3

I'd suggest the same as before: Multiply out the two binomial terms on the left. Note that these have both x and y in them. Once you've done this multiplication and any possible simplification, move the remaining terms on the left to the right side of your equation, leaving just "0=" on the left side. Now see whether you can rearrange these five terms on the right side in descending order by variable y, treating x as a constant.

This may or may not "work," but should prove to be a useful and enlightening first step.