If degree of one polynomial (say $f(x)$) is greater than degree of the other polynomial (say $g(x)$), then is it ever possible for the two polynomials to be equal for $x > 1$?
I was struck with this question when trying to work on a conjecture involving powers. I think I should share some specific values that I got:
Values of $f(x)$:
$4n^3 + 12n^2 + 12n + 4$
$30n^4 + 120n^3 + 180n^2 + 120n + 30$
$12n^5 + 60n^4 + 120n^3 + 120 n^2 + 60n + 12$
Values of $g(x)$:
$n^4 + 2n^3 + n^2$
$6n^5 + 15n^4 + 10n^3 - n$
$2n^6 + 6n^5 + 5n^4 - n^2$
If we compare each corresponding values of $f(x)$ and $g(x)$, we can see that the powers are decreasing for each consecutive term and degree of $f(x)$ is one greater than the degree of $g(x)$. Also they don't seem to be equal for any positive integer $x > 1$.
Of course it is. Here's an example.
$x^4$ and $x^5-3x^2 -4$
Both evaluate to $16$ for $x = 2$.
Post OP's edit
I am not sure what you are trying to convey with the last couple of edits but in one of the comments you said the polynomials had to have strictly positive coefficients.
So, let me modify the above example.
$$P_1(x) = x^5 + x^ 4 + x^3 + x^2 + x + 1$$
$$P_2(x) = 2x^ 4 + x^3 + 4x^2 + x + 5$$
If I managed to avoid typos, $P_1(2) = P_2(2)$.