47.2 Polynomial Division by a Monomial

Polynomial Division by a Monomial is the procedure for dividing a polynomial with two or more terms by a single-term expression, carried out by dividing each term of the polynomial by the monomial individually and then combining the resulting quotients into a single expression. It relies on the principle that a sum divided by a value equals the sum of each addend divided by that same value, which allows the division to be distributed across the dividend's terms rather than requiring the more elaborate long division procedure used when the divisor has multiple terms.

This method produces a quotient directly whenever every term of the dividend is evenly divisible by the monomial divisor, and it produces a remainder or a fractional term when it is not.

Reading the Divisor

Monomial Divisor Factor Reading

Before dividing, the monomial divisor is read as two separate factors: its numerical coefficient and its variable part, including any exponents. For a divisor such as 4x³, the numerical factor is 4 and the variable factor is x³, and both will be applied separately to each term of the dividend.

Separating and Dividing Term by Term

Dividend Term-by-Term Separation

The dividend is split into its individual terms, each of which will be divided by the monomial divisor on its own, following the distributive property of division over addition.

Coefficient Quotient Calculation

For each term, the numerical coefficient of the dividend term is divided by the numerical coefficient of the divisor using ordinary arithmetic, producing the coefficient of that quotient term.

Common-Base Exponent Subtraction

The variable part of each quotient term is found by subtracting the divisor's exponent from the dividend term's exponent for any variable that appears in both, following the quotient rule of exponents.

Quotient Term Sign Determination

The sign of each quotient term follows the same sign rules as ordinary division: a positive dividend term divided by a positive divisor gives a positive quotient term, while a negative dividend term divided by a positive divisor gives a negative quotient term.

Checking and Assembling the Result

Each-Term Exact Divisibility Check

Each term of the dividend is checked individually to confirm whether the divisor's coefficient evenly divides the term's coefficient and whether the divisor's variable exponents are less than or equal to the corresponding exponents in the term; if either condition fails for a given term, that term contributes a fractional expression to the quotient rather than a whole-number, whole-exponent term.

Monomial-Division Quotient Assembly

Once every term of the dividend has been divided individually, the resulting quotient terms are combined by addition or subtraction, preserving whatever sign each term carried, to form the complete quotient expression.

Monomial-Division Result Standardization

The assembled quotient is arranged in standard form, ordering its terms from highest degree to lowest degree, matching the conventional presentation used for any simplified polynomial expression.

Worked Example

Dividing a Trinomial by a Monomial

Consider dividing 12x⁴ + 8x³ − 6x² by the monomial 2x². Each term is divided separately: the coefficients 12, 8, and −6 are each divided by 2, giving 6, 4, and −3; the exponents 4, 3, and 2 each have 2 subtracted, giving 2, 1, and 0.

The final term's exponent, 2 − 2 = 0, reduces x⁰ to 1, leaving a plain constant term of −3 in the assembled quotient.

Visual Representation of Term Distribution