are the two polynomials the same? If you have three polynomials $f, g$ and $h$. And the following is true:$\int^{1}_{-1}f\cdot g=\int^{1}_{-1}f\cdot h$. Does this directly imply that $h=g$ (assuming $f$ is not the zero polynomial)? I think it does imply, but I am not sure. Thanks.
Let $f(x)=x$, $g(x)=1$ and $h(x)=1+x^2$. Then $$\int_{-1}^1 f(g-h) = -\int_{-1}^1 x^3\, dx =0.$$
Two diff polynomials can yield same integral value(we can also say it as grazed same area)
in same given limits(a,b) in this case (-1,1)

it doesnt mean both polynomials are same.

let f.g=k,f.h=l

and integral k in -1 to 1 =integral l in -1 to 1.

it doesnt mean k and l are same.