47.4 Polynomial Long Division Cycle

Polynomial Long Division Cycle is the repeated sequence of steps, dividing, multiplying, subtracting, and bringing down, that transforms a prepared dividend and divisor into a completed quotient and remainder. Once the dividend and divisor have been ordered and gap-filled during preparation, this cycle is applied over and over, once for each degree position, until no further division is possible, producing one term of the quotient with each pass through the cycle.

Each pass of the cycle reduces the degree of the remaining dividend by at least one, guaranteeing that the process eventually terminates rather than continuing indefinitely.

The First Step of Each Pass

Leading-Term Quotient Calculation

At the start of each pass, the leading term of the current dividend, or intermediate remainder from a previous pass, is divided by the leading term of the divisor, following the quotient rule of exponents and ordinary coefficient division, to produce the next term of the quotient.

Quotient Term Placement

This newly calculated quotient term is written in the quotient position set aside during preparation, aligned according to its own degree, above the term of the dividend that shares that same degree.

Forming and Subtracting the Partial Product

Divisor-by-Quotient-Term Product

The entire divisor is then multiplied by the quotient term just found, producing a partial product that will be subtracted from the current dividend.

Long-Division Product Degree Alignment

Because the quotient term was chosen specifically so that its product with the divisor's leading term matches the degree of the current dividend's leading term, the partial product aligns directly under the corresponding terms of the current dividend, ensuring the subtraction that follows cancels the leading term exactly.

Complete Partial Product Subtraction

The partial product is subtracted from the current dividend across all of its terms, not merely the leading term, since the divisor typically has more than one term and each of its terms contributes to the partial product.

Continuing the Cycle

Intermediate Polynomial Remainder

The result of this subtraction is an intermediate remainder, a polynomial whose degree is strictly lower than the degree of the current dividend that produced it, since the leading terms were designed to cancel exactly.

Next Dividend Term Incorporation

If any terms of the original dividend at lower degrees have not yet been included in the subtraction, they are incorporated into the intermediate remainder at this stage, effectively becoming the working dividend for the next pass of the cycle.

Repeated Leading-Term Division

With the intermediate remainder now serving as the current dividend, the entire cycle repeats from its first step: dividing the new leading term by the divisor's leading term to generate the next quotient term, and continuing through multiplication and subtraction once again.

Ending the Cycle

Long-Division Termination Condition

The cycle terminates when the degree of the current intermediate remainder becomes less than the degree of the divisor, since at that point the divisor's leading term can no longer divide evenly into the remainder's leading term to produce a valid polynomial quotient term. At that point, the final intermediate remainder becomes the division's overall remainder, and the accumulated quotient terms together form the complete quotient.