Polynomial division
Enter coefficients in descending order, including zeros for missing powers. In expression mode enter both polynomials; coefficient controls are used only in coefficient mode. Powers up to 64 are supported. The slant asymptote exists when the numerator degree exceeds the denominator degree by one. Division writes f = q + r/d. This calculator also reports a higher-degree quotient when applicable, without calling it a straight-line asymptote.
Why your number may differ
For the defaults, (x³ + 2x + 1)/(x² + 1) has asymptote y = x and remainder x + 1. Its real zero is approximately −0.453398: a rational-root-only search misses it. We isolate real roots numerically through derivative intervals. A common factor creates a hole only if the numerator cancels the entire denominator multiplicity. Graphs break at excluded points; finite sampling and numerical tolerances still limit precision. The curve can cross its asymptote. See OpenStax.