47 Polynomial Division

Polynomial Division is the study of dividing one polynomial by another, extending division from simple fractions to expressions involving variables, using term-by-term division for a monomial divisor and a systematic long-division cycle for a polynomial divisor with more than one term.

The Scope of Polynomial Division

Dividing a polynomial by another polynomial produces a quotient, and possibly a remainder when the division does not come out evenly, mirroring the structure of ordinary long division of whole numbers. The method used depends on whether the divisor is a single monomial or a polynomial with two or more terms, since these two cases require substantially different procedures.

Dividing a Polynomial by a Monomial

When the divisor is a single monomial, division is performed by dividing each term of the polynomial by the monomial separately, applying the quotient rule for exponents to each variable factor.

Each term's coefficient is divided by the monomial's coefficient using ordinary division, and each term's variable factor is divided by the monomial's variable factor by subtracting exponents, exactly as this term-by-term approach mirrors the distributive property used in monomial-by-polynomial multiplication, applied here in reverse.

Preparing for Polynomial Long Division

When the divisor has more than one term, polynomial long division is required, and both the dividend and divisor must first be written in standard form, arranged by descending degree. If any degree between the highest and lowest is missing from the dividend, a placeholder term with a coefficient of zero should be inserted for that missing degree, ensuring that terms of matching degree remain properly aligned throughout the division process, exactly as zero placeholders are used in long division of numbers with internal gaps.

The Long Division Cycle

Polynomial long division proceeds through a repeating four-step cycle, applied once per term of the resulting quotient: divide the leading term of the current remainder by the leading term of the divisor to obtain the next quotient term; multiply the entire divisor by that new quotient term; subtract this product from the current remainder, changing every sign during the subtraction; and bring down the next term of the original dividend to form the new current remainder. This cycle repeats until the current remainder's degree is smaller than the divisor's degree, at which point the division is complete.

Expressing the Quotient and Remainder

When a polynomial division leaves no remainder, the result is written simply as the quotient polynomial, and the divisor is confirmed to be an exact factor of the original dividend. When a nonzero remainder is left over, the complete result is written as the quotient plus the remainder expressed as a fraction over the original divisor:

This final remainder term always has a degree strictly less than the degree of the divisor, exactly as a numerical long division's remainder is always smaller than its divisor.

Degree Considerations in Polynomial Division

The degree of the quotient in a polynomial long division is always equal to the degree of the dividend minus the degree of the divisor, a relationship that can be used to predict how many terms the quotient should contain before beginning the division. If the dividend's degree is smaller than the divisor's degree from the outset, the division cannot proceed at all in the usual sense, and the entire dividend is treated as the remainder, with a quotient of zero.

Verifying a Polynomial Division

A completed polynomial division is verified using the same relationship that governs ordinary numerical division: multiplying the quotient by the divisor and adding the remainder should reproduce the original dividend exactly.

This check, performed through polynomial multiplication and addition, both confirms the correctness of the division and reinforces the inverse relationship between polynomial multiplication and polynomial division.

Diagnosing Errors in Polynomial Division

Common errors in this area include forgetting to change every sign during the subtraction step of the long division cycle, omitting a zero-coefficient placeholder for a missing degree in the dividend and thereby misaligning subsequent terms, stopping the division cycle too early or too late relative to when the current remainder's degree drops below the divisor's degree, and reporting only the quotient while forgetting to include a nonzero remainder as part of the complete division result.