Polynomial division

An algebraic fraction is called improper when the degree of the numerator is greater than or equal to that of the denominator. In that case it can always be rewritten as the sum of a polynomial (the quotient) and a proper fraction, whose numerator is the remainder of the division:

$$\frac{N(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}, \qquad \deg R < \deg D.$$

The tool performs this division keeping the coefficients as exact rational numbers, with no rounding. A literal parameter a is allowed (linear in the values you type): the only constraint is that the leading coefficient of the denominator be a number, not an expression containing a.

How to use the tool

• Coefficient boxes. In each box enter an integer, a fraction (for example 3/4) or a linear expression in the parameter (for example 2a-1). The sign goes inside the box.
• Maximum degree. The selector sets the maximum degree, the same for numerator and denominator. Changing it clears the coefficients.
• Use the monic denominator. Divides the denominator by its own leading coefficient; the result is shown by factoring out $\frac{1}{k}$.
• Random values. Generates a purely numerical, ready-to-run division.
• Quick result. Shows the complete table with quotient and remainder highlighted, without the explanation.
• Step-by-step solution. Rebuilds the division one term at a time, with the long-division layout and an explicit vertical subtraction.
• Analyse the parameter a. When the parameter is present, it reports for which values of a the remainder vanishes and for which its degree drops.