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This is a variation of this question, but in my case I want to know if two polynomials of different degrees and non-negative coefficients can have more than one intersection in the positive $x$-axis.

For example, this is the plot of the polynomials $15 x^4$ and $45 x^3 + 105 x^2 + 85 x + 25$.

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Of course they can. Say $3x$ and $x^2+2$. They intersect in $1$ and $2$:

$$ 3x=x^2+2\implies x^2-3x+2=0\implies (x-2)(x-1)=0$$

nonuser
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  • @user126444: You can get any number of intersections at any values of $x$ that you want!. Just find a polynomal that has roots at those $x$-values (like greedoid did for $x=1$ and $x=2$), then move the negative coefficients to the other side (that originally had the zero, just as greedoid did with the $-3x$ term). – Ingix Oct 18 '18 at 21:14