problem: Let $h$ be a polynomial of degree $5$, and suppose that $h(x) = (x^2-7x+10) \cdot g(x)$, where $g(x)$ is a polynomial of degree $b$.  Find $b$.
solution: Recall that for positive integers $a$ and $b$ $x^a x^b = x^{a+b}$.  Thus, if $h(x)$, $f(x)$, and $g(x)$ are polynomials such that $h(x)=f(x) \cdot g(x)$ and if $ x^a $ and $ x^b $ are the highest degree terms of $f(x)$ and $g(x)$ then the highest degree term of $h(x)$ will be $x^{a+b}$.  In our case $f(x) = x^2-7x+10$, the highest degree term is $x^2$ and $h(x)$ has highest degree term $x^5$.  Thus, solving for $b$ in the equation $x^5 = x^2 x^b = x^{2+b}$, we find $b=\boxed{3}$.