Is it possible to use repeated synthetic division (rather than long division) to find a slant asymptote for a rational function such as $\displaystyle \frac{2x^3 + 3x^2 + 5x + 7}{(x-1)(x-3)}$? It appears to work, but I am not sure that it is valid to ignore the remainder term from the first synthetic division.
2026-04-01 20:01:02.1775073662
Finding Slant Asymptotes using synthetic division rather than long division
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It's OK, for a slant asymptote to exist degree of numerator must exceed degree of denominator by $1$. In this case the quotient will be a polynomial $f(x)=ax+b$, and $y=f(x)$ is the asymptote.
The remainder in the first division is completely immaterial, as you can see by substituting an arbitrary constant $c$ instead of $7$ in the numerator.